On Spectral Properties of the L Operator in the Lax Pair of the sine-Gordon Equation
We analyze the periodic spectrum of the L operator in the Lax pair of the sine-Gordon equation in terms of the regularity of the potential.
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irk-123456789-1458822019-02-03T01:23:21Z On Spectral Properties of the L Operator in the Lax Pair of the sine-Gordon Equation Kappeler, T. Widmer, Y. We analyze the periodic spectrum of the L operator in the Lax pair of the sine-Gordon equation in terms of the regularity of the potential. Ми аналiзуємо перiодичний спектр оператора L у парi Лакса рiвняння sine-Гордона в термiнах регулярностi потенцiалу. 2018 Article On Spectral Properties of the L Operator in the Lax Pair of the sine-Gordon Equation / T. Kappeler, Y. Widmer // Журнал математической физики, анализа, геометрии. — 2018. — Т. 14, № 4. — С. 452-509. — Бібліогр.: 9 назв. — англ. 1812-9471 DOI: https://doi.org/10.15407/mag14.04.452 Mathematics Subject Classification 2000: 37K10, 35Q55 http://dspace.nbuv.gov.ua/handle/123456789/145882 en Журнал математической физики, анализа, геометрии Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
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We analyze the periodic spectrum of the L operator in the Lax pair of the sine-Gordon equation in terms of the regularity of the potential. |
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Article |
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Kappeler, T. Widmer, Y. |
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Kappeler, T. Widmer, Y. On Spectral Properties of the L Operator in the Lax Pair of the sine-Gordon Equation Журнал математической физики, анализа, геометрии |
| author_facet |
Kappeler, T. Widmer, Y. |
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Kappeler, T. |
| title |
On Spectral Properties of the L Operator in the Lax Pair of the sine-Gordon Equation |
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On Spectral Properties of the L Operator in the Lax Pair of the sine-Gordon Equation |
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On Spectral Properties of the L Operator in the Lax Pair of the sine-Gordon Equation |
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On Spectral Properties of the L Operator in the Lax Pair of the sine-Gordon Equation |
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On Spectral Properties of the L Operator in the Lax Pair of the sine-Gordon Equation |
| title_sort |
on spectral properties of the l operator in the lax pair of the sine-gordon equation |
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Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
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2018 |
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http://dspace.nbuv.gov.ua/handle/123456789/145882 |
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On Spectral Properties of the L Operator in the Lax Pair of the sine-Gordon Equation / T. Kappeler, Y. Widmer // Журнал математической физики, анализа, геометрии. — 2018. — Т. 14, № 4. — С. 452-509. — Бібліогр.: 9 назв. — англ. |
| series |
Журнал математической физики, анализа, геометрии |
| work_keys_str_mv |
AT kappelert onspectralpropertiesoftheloperatorinthelaxpairofthesinegordonequation AT widmery onspectralpropertiesoftheloperatorinthelaxpairofthesinegordonequation |
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2025-07-10T22:47:47Z |
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2025-07-10T22:47:47Z |
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1837301969761337344 |
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Journal of Mathematical Physics, Analysis, Geometry
2018, Vol. 14, No. 4, pp. 452–509
doi: https://doi.org/10.15407/mag14.04.452
On Spectral Properties of the L Operator in
the Lax Pair of the sine-Gordon Equation
Thomas Kappeler and Yannick Widmer
To Professor Vladimir Aleksandrovich Marchenko
on the occasion of his ninety fifth birthday
We analyze the periodic spectrum of the L operator in the Lax pair of
the sine-Gordon equation in terms of the regularity of the potential.
Key words: sine-Gordon equation, sinh-Gordon equation, Lax pair, de-
cay properties of gap lengths.
Mathematical Subject Classification 2010: 37K10, 35Q55.
1. Introduction
We consider the complex sine-Gordon equation
utt − uxx = − sinhu, x ∈ T = R/Z, t ∈ R, (1.1)
where u is assumed to be complex valued. In case u is real valued, (1.1) is referred
to as the sinh-Gordon equation and in case u is purely imaginary, as real sine-
Gordon equation. Equation (1.1) is a nonlinear perturbation of the (complex)
Klein–Gordon equation utt − uxx = mu (with m = −1). It has wide ranging ap-
plications in geometry and quantum mechanics and has been extensively studied,
although most of the work has been done for its version in light cone coordinates
and hence does not apply to the periodic in space setup of (1.1). According to [2]
(cf. also [8]), it admits a Lax pair. To describe it, note that (1.1) can be written
as a system for (u1, u2) := (u, ut)(
u1
u2
)
t
=
(
u2
u1xx − sinhu1
)
. (1.2)
In order to work with function spaces consisting of pairs of functions of equal
regularity, we introduce
(q, p) := (u1,−P−1u2) ∈ H1
c , (1.3)
where for any s, Hs
c denotes the Sobolev space Hs(T,C)×Hs(T,C) (≡ Hs(T,C2))
and P the Fourier multiplier operator P :=
√
1− ∂2
x. When expressed in these
c© Thomas Kappeler and Yannick Widmer, 2018
https://doi.org/10.15407/mag14.04.452
On Spectral Properties of the L Operator. . . 453
coordinates, equation (1.2) becomes(
qt
pt
)
=
(
−Pp
Pq + P−1(sinh(q)− q)
)
.
For any v = (q, p) ∈ H1
c , define the following differential operators:
Q(v) = Q1∂x +Q0(v), K(v) = K1∂x +K0(v), (1.4)
where the coefficients Q1, Q0, K1, K0 are the 4× 4 matrices given by
Q1 =
(
−J
)
, Q0(v) =
(
A(v) B(v)
B(v)
)
,
K1 =
(
−I
I
)
, K0(v) =
(
−2JB(v)
−2B(v)J
)
,
with I, J , R, Z denoting the 2× 2 matrices
I =
(
1
1
)
, J =
(
1
−1
)
, R =
(
i
−i
)
, Z =
(
1
1
)
(1.5)
and
A(v) := −1
4
(Pp+ qx)Z, B(v) :=
1
4
(
exp(−q/2)
exp(q/2)
)
. (1.6)
Here and in the sequel, we suppress matrix coefficients which vanish, so, e.g.,(
1
−1
)
stands for
(
0 1
−1 0
)
. One verifies that t 7→ v(t) is a solution of (1.1) iff
t 7→ (Q(v(t)),K(v(t))) satisfies
Qt = [K,Q]. (1.7)
The pair of operators Q,K is referred to as Lax pair for the sine-Gordon equation
and Q as the corresponding L operator. Note that (1.7) leads to an abundance
of first integrals of (1.1): expressed in a somewhat informal way (i.e., without
addressing issues of regularity) it follows from (1.7) that for any solution t 7→
v(t) of (1.1), the periodic spectrum specperQ(v(t)) of the operator Q(v(t)) is
independent of t. Here for any v ∈ H1
c , the periodic spectrum of Q(v) is the
spectrum of the operatorQ(v), considered with the Sobolev spaceH1(R/(2Z),C2)
of two periodic functions with values in C2 as its domain. Since for any v ∈ H1
c ,
specperQ(v) is discrete this is saying that the periodic eigenvalues of Q(v) are
first integrals for the sine-Gordon equation.
Besides being of interest in its own right, the periodic spectrum of the operator
Q(v) and its associated spectral curve play a very important role in the construc-
tion of the normal form of the sine-Gordon equation. With this application in
mind, the aim of this paper is to analyze specperQ(v).
454 Thomas Kappeler and Yannick Widmer
To state our main results, we first need to introduce some more notation.
Define the domains D0 :=
{
z ∈ C : |z − 1
4 | <
1
4π
}
and for any n ≥ 1,
Dn := {λ ∈ C : |λ− nπ| < π/3} , D−n :=
{
λ ∈ C :
1
16λ
∈ Dn
}
.
Furthermore, let B0 := {λ ∈ C : |λ| ≤ π/2} and for any n ≥ 1
Bn := {λ ∈ C : |λ| < nπ + π/2} , B−n :=
{
λ ∈ C : |λ| ≤ 1
16(nπ + π/2)
}
,
and denote by An the open annulus An := Bn \ B−n. By the Counting Lemma
(cf. Lemma 3.11 in Section 3) for any potential in H1
c , there exist a neighborhood
U in H1
c and an integer N ≥ 1, such that for any v ∈ U and any n > N , the
operator Q(v) has exactly two periodic eigenvalues in each of the domains Dn,
−Dn, D−n, and −D−n and exactly 4+8N roots in the annulus AN , counted with
their multiplicities. There are no further eigenvalues. We denote the eigenvalues
in Dk, |k| ≥ N by λ+
k , λ
−
k and list them in the following order, [ |λ−k | < |λ
+
k | ]
or [ |λ−k | = |λ+
k | and Imλ−k ≤ Imλ+
k ]. By symmetry, for any k ∈ Z with |k| >
N , the two eigenvalues in −Dk are given by −λ+
k and −λ−k . By a slight abuse of
terminology, we refer to the difference λ+
k − λ
−
k as the k’th gap length.
Theorem 1.1. For any v ∈ Hs+1
c with s ≥ 0 and for N ≥ 1 given as above,
the following asymptotic estimates of the gap lengths of the periodic spectrum
Q(v) hold: there exists C > 0 so that∑
n>N
〈n〉2s
∣∣λ+
n − λ−n
∣∣2 ≤ C, (1.8)∑
n>N
〈n〉2s
∣∣(16λ+
−n)−1 − (16λ−−n)−1
∣∣2 ≤ C. (1.9)
Furthermore, N and C can be chosen locally uniformly with respect to v.
To state the second result, we first need to introduce some more notation.
Let v ∈ H1
c and let N ≥ 1 be as above. Then for any N ′ ≥ N, v ∈ H1
c is said to
be a right [left] sided N ′-gap potential if
∀n > N ′ λ+
n = λ−n [∀n > N ′ λ+
−n = λ−−n].
It is said to be a right [left] sided finite gap potential if it is a right [left] sided
N ′-gap potential for some N ′ ≥ N . For any s ≥ 1, denote by LFGsc and RFGsc
the following subsets of Hs
c :
LFGsc := {v ∈ Hs
c : v left sided finite gap potential}
and
RFGsc := {v ∈ Hs
c : v right sided finite gap potential} .
Theorem 1.2. For any real number s with s ≥ 1, the sets LFGsc and RFGsc
are dense in Hs
c .
On Spectral Properties of the L Operator. . . 455
Comments:
(i) The presented results are part of a larger project of constructing normal coor-
dinates, also referred to as Birkhoff coordinates, for the sine-Gordon equation.
When expressed in these coordinates, the sine-Gordon equation is in normal
form. It allows to solve it by quadrature and to obtain KAM type theo-
rems for (small) perturbations of it. Previously, such a program has been
carried out for the KdV equation (cf. [5] and references therein) and the
defocusing NLS equation (cf. [3] and references therein). To carry this out
for the sine-Gordon equation turns out to be more challenging since in this
case, the spectral analysis of the L-operator (cf. 1.4) is more complicated (cf.
Sections 2 and 3).
(ii) More detailed versions of Theorem 1.1 and Theorem 1.2 are stated in The-
orem 4.10 and Theorem 4.15 in Section 4. Their proofs are based on a
Lyapunov–Schmidt decomposition developed in previous work for the Hill
and Zakharov–Shabat operators — see [1, 4, 6, 9] and references therein. Our
work confirms that this method of proof can be applied in a wide variety of
cases, but that the estimates needed can become quite involved.
(iii)Many questions remain open which we plan to address in future work. We ex-
pect that the developed techniques allow to obtain also asymptotic estimates
for µn − (λ+
n + λ−n )/2, similar to the ones in (1.8) and (1.9) obtained for the
gap lengths. Here µn denote the Dirichlet eigenvalues of Q(v) (cf. Section 3).
Furthermore, we expect that λ+
n , λ
−
n admit asymptotic expansions in n−1 of
the type Marchenko proved in seminal work for the periodic eigenvalues of
Hill’s operator and of the Zhakarov–Shabat operator (cf. [7]) and that (1.8),
(1.9) characterize the regularity of the potential v in case v is real or purely
complex valued. It means that in such a case, any v ∈ H1
c satisfying (1.8)
and (1.9) is actually in Hs+1
c . Although the statement of Theorem 1.2 is suf-
ficient for the applications we have in mind we expect that with the methods
developed, it can be shown that for any s ≥ 1, the intersection LFGsc∩RFGsc
is dense in Hs
c .
Related work: As already mentioned, the sine-Gordon equation has been
extensively studied, although most of the work has been done for its version in
light cone coordinates. We only mention that in the periodic in space setup,
important contributions can be found in [8] where parts of the material discussed
in Sections 2 and 3 are presented and the spectral curve associated with Q(v) is
studied.
Organisation: In Section 2, we record the results needed on the fundamental
solution of the operator Q(v), the main purpose being to introduce notation
and to state the results in the form needed later. In Section 3, we study the
asymptotics of the periodic, the Dirichlet, and the Neumann eigenvalues of the
operator Q(v) for v ∈ H1
c . They provide the setup for the proofs of Theorem 1.1
and Theorem 1.2 in Section 4. Although the material in these two sections is by
and large known (cf., e.g., [8]), for the convenience of the reader, we included the
456 Thomas Kappeler and Yannick Widmer
proofs of the stated results. In Section 4 we prove Theorem 1.1 and Theorem 1.2.
2. Fundamental solution
In this section we study the fundamental solution of the differential operator
Q = Q(v) defined in (1.4) where v = (q, p) ∈ H1
c . The results obtained will be
used in particular in Section 3 in order to analyze the periodic and the Dirichlet
spectrum of Q. Note that QF = 0 if and only if F = 0 and for any given λ ∈
C∗ = C \ {0}, a function F in H1
loc(R,C4) is a solution of
QF = λF (2.1)
if and only if F = (f, λ−1Bf) and f satisfies the following first order ODE:
− J∂xf + (A+B2/λ)f = λf. (2.2)
Here and in the sequel we often write ∂xf for fx. Let M = M(x, λ, v) ∈ C2×2 be
the fundamental solution for equation (2.2), meaning that
(−J∂x +A(x) +B2(x)/λ)M(x, λ, v) = λM(x, λ, v), M(0, λ, v) = Id.
Clearly one has
∂xM = J
(
λ−A−B2/λ
)
M, (2.3)
or, taking the definition (1.6) into account,
∂xM(x, λ, v) = J
(
λ+
1
4
(Pp(x) + qx(x))
(
1
1
)
− 1
16λ
(
e−q(x)
eq(x)
))
M.
It is convenient to also use the matrix valued function M instead of M where
M(x, λ, v) = TM(x, λ, v)T−1, T =
(
1 i
1 −i
)
, T−1 =
1
2
(
1 1
−i i
)
. (2.4)
Introduce
Q =
(
T
T
)
Q
(
T−1
T−1
)
= Q1∂x +Q0(v), (2.5)
where
Q1 =
(
R
)
, Q0(v) =
(
A(v) B(v)
B(v)
)
, (2.6)
A(v) = TA(v)T−1 = − i
4
(Pp+ qx)J, (2.7)
B(v) = TB(v)T−1 =
1
4
(
cosh(q/2) − sinh(q/2)
− sinh(q/2) cosh(q/2)
)
, (2.8)
and hence
B(v)2 = TB(v)2T−1 =
1
16
(
cosh(q) − sinh(q)
− sinh(q) cosh(q)
)
. (2.9)
On Spectral Properties of the L Operator. . . 457
An element F in H1
loc(R,C4) is a solution of QF = λF iff QTF = λTF . Further-
more F on H1
loc(R,C4) is a solution of
QF = λF (2.10)
iff F = (F1, λ
−1BF1) and
∂xF1 = −R
(
λ−A− B2/λ
)
F1. (2.11)
Hence M satisfies
∂xM = −R
(
λ−A− B2/λ
)
M, M(0, λ, v) = I. (2.12)
First we discuss symmetries of the fundamental solution M needed in the sequel.
Proposition 2.1 (Symmetries). For any (x, λ, v) ∈ [0,∞)× C∗ ×H1
c
(i) (Reflection of λ)
M(x,−λ, v) = −RM(x, λ, v)R and M(x,−λ, v) = −ZM(x, λ, v)Z.
(ii) (Reciprocity of λ)
M
(
x,
1
16λ
, q, p
)
= −Re−iRq(x)/2M(x, λ,−q, p)eiRq(0)/2R
or
M (x, λ,−q, p) = −ReiRq(x)/2M
(
x,
1
16λ
, q, p
)
e−iRq(0)/2R.
(iii)(Conjugation)
M(x, λ, v) = M(x, λ, v) and M(x, λ, v) = ZM(x, λ, v)Z.
(iv) (Reflection of v)
M(x, λ,−v) = JM(x, λ, v)J−1 and M(x, λ,−v) = RM(x, λ, v)R−1.
Proof. (i) First note that M(x,−λ, v) and −RM(x, λ, v)R coincide at x = 0.
Hence it suffices to show that they satisfy both the same first order differential
equation. By (2.3) M(x,−λ, v) satisfies the equation
∂xM(x,−λ, v) = J
(
−λ+
1
4
(Pp(x) + qx(x))Z +
1
16λ
eiRq(x)
)
M(x,−λ, v)
and thus
∂x (−RM(x, λ, v)R)
458 Thomas Kappeler and Yannick Widmer
= −RJ
(
λ+
1
4
(Pp(x) + qx(x))Z − 1
16λ
eiRq(x)
)
M(x, λ, v)R.
Since RJ = −JR, RZ = −ZR and ReiRq = eiRqR, one concludes that
−RM(x, λ, v)R satisfies the same equation as M(x,−λ, v),
∂x (−RM(x, λ, v)R)
= J
(
−λ+
1
4
(Pp(x) + qx(x))Z +
1
16λ
eiRq(x)
)
(−RM(x, λ, v)R)
as claimed. Concerning the second identity of item (i), note that
Z = −iTRT−1 (2.13)
and hence −ZMZ = −ZTMT−1Z = TRMRT−1 which yields the claimed iden-
tity for M(x,−λ, v).
(ii) Again note that M(x,− 1
16λ , q, p) and e−iRq(x)/2M(x, λ,−q, p)eiRq(0)/2 co-
incide at x = 0. Hence it suffices to show that they satisfy both the same first
order differential equation. By (2.3), M(x,− 1
16λ , q, p) satisfies
∂x
(
x,− 1
16λ
, q, p
)
= J
(
− 1
16λ
+
1
4
(Pp(x) + qx(x))Z + λeiRq(x)
)
M
(
x,− 1
16λ
, q, p
)
. (2.14)
On the other hand, ∂x
(
e−iRq(x)/2M(x, λ,−q, p)
)
equals
e−iRq(x)/2J
(
λ+
1
4
(Pp(x)− qx(x))Z − 1
16λ
e−iRq(x)
)
M(x, λ,−q, p)
− 1
2
qx(x)iRe−iRq(x)/2M(x, λ,−q, p).
Since e−iRq/2J = JeiRq/2, eiRq/2Z = Ze−iRq/2 and iR = −JZ, one gets
∂x
(
e−iRq(x)/2M(x, λ,−q, p)
)
= J
(
λeiRq(x) +
1
4
(Pp(x) + qx(x))Z − 1
16λ
)
e−iRq(x)/2M(x, λ,−q, p). (2.15)
By the latter identity and (2.14), one sees that M
(
x,− 1
16λ , q, p
)
and
e−iRq(x)/2M(x, λ,−q, p)eiRq(0)/2 satisfy the same differential equation and hence
must coincide. The first identity of item (ii) then follows from (i). The second
identity of (ii) is obtained using the first one.
(iii) Take the complex conjugate of (2.3), ∂xM(x, λ, v) = J
(
λ − A(v) −
B(v)2/λ
)
M(x, λ, v). Since A(v) = A(v), B(v)2 = B(v)2 and M(0, λ, v) = I,
one concludes that M(x, λ, v) = M(x, λ, v). The second identity of item (iii) then
also follows since by (2.4), T−1 = T−1Z.
On Spectral Properties of the L Operator. . . 459
(iv) Note that M(x, λ,−v) satisfies
∂xM(x, λ,−v) = J
(
λ− 1
4
(Pp(x) + qx(x))Z − 1
16λ
e−iRq(x)
)
M(x, λ,−v) (2.16)
while ∂xJM(x, λ, v) = JJ
(
λ+ 1
4(Pp(x) + qx(x))Z − 1
16λe
iRq(x)
)
M(x, λ, v). The
first claimed identity then follows from JZ = −ZJ and JeiRq = e−iRqJ and the
second one from TJT−1 = −R.
To obtain estimates for the fundamental solution of Q, we write (2.12) as an
integral equation and represent its solution as a series. Here and in the sequel
we will use the Euclidean norm for vectors in C2 and the induced operator norm
for matrices. We write M =
(
m1 m2
m3 m4
)
where mj = mj(x, λ, v). The norm |M |
of M induced by the Hermitian norm on C2 can be bounded as follows:
|M | ≤ 2 max{|m1|+ |m2|, |m3|+ |m4|}. (2.17)
Let us first compute M(x, λ, 0). By (2.12), it satisfies ∂xM = −R
(
λ− 1
16λ
)
M.
For λ ∈ C∗, let
Eω(x) := e−Rωx =
(
e−iωx
eiωx
)
, ω ≡ ω(λ) = λ− 1
16λ
. (2.18)
Then ∂xEω = −RωEω and hence Eω(x) =M(x, λ, 0). Note that ω(λ) = 0 iff λ =
±1
4 and
ω
(
1
16λ
)
= ω(−λ), ω(−λ) = −ω(λ), ∀λ ∈ C∗. (2.19)
For v ∈ H1
c , M(x) − Eω(x) = Eω(0)M(x) − Eω(x)M(0) satisfies the integral
equation
M(x)− Eω(x) =
∫ x
0
∂s (Eω(x− s)M(s)) ds
=
∫ x
0
RωEω(x− s)M(s)− Eω(x− s)R
(
λ−A− B2/λ
)
M(s) ds
=
∫ x
0
Eω(x− s)R
(
ω − λ+A+ B2/λ
)
M(s) ds
or
M(x)− Eω(x) =
∫ x
0
Eω(x− s)Φλ(s)M(s)ds, (2.20)
where by (2.7)–(2.9)
Φλ(s) = R(ω − λ+A+ B2/λ)
= R
(
1
16λ
(
cosh(q)− 1 − sinh(q)
− sinh(q) cosh(q)− 1
)
− i
4
(Pp+ qx)J
)
.
460 Thomas Kappeler and Yannick Widmer
To investigate the regularity of the fundamental solution we use equation (2.20)
to find a series representation for M. Let M0(x) = Eω(x) and define Mn+1(x)
inductively by
Mn+1(x) =
∫ x
0
Eω(x− s)Φλ(s)Mn(s) ds. (2.21)
Using that Eω(x−s) = Eω(x)Eω(−s), one obtains the following identity forMn+1,
Mn+1(x) =
∫ x
0
Eω(x− s)Φλ(s)Mn(s) ds
= Eω(x)
∫
0≤x1≤···≤xn≤x
Eω(−xn)Φλ(xn)Eω(xn) · · ·
× Eω(−x1)Φλ(x1)Eω(x1) dx1 · · · dxn.
As usual, we denote by ‖q‖s the Hs(T,C)-norm of q =
∑
k∈Z q̂2ke
2πikx
‖q‖s =
(∑
k∈Z
|q̂2k|2〈2k〉2s
)1/2
, 〈2k〉 =
√
1 + (2kπ)2 (2.22)
and write ‖v‖s = ‖q‖s+‖p‖s as well as ‖v‖0 ≡ ‖v‖L2 = ‖q‖L2 +‖p‖L2 . Note that
for any q ∈ H1
C, one has
‖q‖L∞ ≤ 2 ‖q‖1 . (2.23)
Theorem 2.2 (Regularity of the fundamental solution). The series M(x) =∑∞
n=0Mn(x), with Mn(x) given by (2.21), converges in C2×2 absolutely, uni-
formly on bounded closed subsets of [0,∞) × C∗ × H1
c . M is continuous in
x, λ, v and analytic in v = (q, p) and λ as a map with values in the Ba-
nach space C([0, 2],C2×2) of continuous functions with values in C2×2, en-
dowed with the supremum norm. It is the unique solution of (2.12), ∂xM =
−R
(
λ−A− B2/λ
)
M, M(0, λ, v) = I, implying that M and ∂xM are analytic
in v and λ as maps with values in L2
(
[0, 2],C2×2
)
. Furthermore M satisfies the
following estimate for any 0 ≤ x ≤ 2, λ ∈ C∗, v ∈ H1
c ,
|M(x, λ, v)| ≤ e| Imω|x exp
(
e2| Imω|x
(
x
|λ|
e2‖q‖1 +
√
x ‖v‖1
))
.
Proof. Clearly, one has |M0(x)| = |Eω(x)| = e|Imω|x. To estimate Mn+1 for
n ≥ 0, we first need to estimate F (x, λ, v) :=
∫ x
0 Eω(−s)Φλ(s)Eω(s)ds. Use the
bound (2.17) of the matrix norm and the identity
Eω(−s)Φλ(s)Eω(s) =
i
16λ
(
cosh(q(s))− 1 − sinh(q(s))e2iωs
sinh(q(s))e−2iωs 1− cosh(q(s))
)
+
1
4
(Pp(s) + qx(s))
(
e2iωs
e−2iωs
)
(2.24)
to conclude that
|F (x, λ, v)| ≤ 1
|λ|
max
±
∫ x
0
(
| cosh(q(s))− 1|+ |e±2iωs sinh(q(s))|
)
ds
On Spectral Properties of the L Operator. . . 461
+ max
±
∫ x
0
∣∣(Pp(s) + qx(s))e±2iωs
∣∣ds.
Since
| cosh(q)− 1| ≤
∑
n≥1
1
(2n)!
|q|2n and | sinh(q)| ≤
∑
n≥0
1
(2n+ 1)!
|q|2n+1,
one has
| cosh(q)− 1|+ | sinh(q)| ≤ e|q| − 1.
Using that
|e±2iωs| ≤ e2| Imω|s and e2| Imω|s ≥ 1,
it then follows that
|F (x, λ, v)| ≤
∫ x
0
e2| Imω|s
(
e|q(s)|
1
|λ|
+ |Pp(s) + qx(s)|
)
ds.
Combining (2.23) with the estimate∥∥∥e2| Imω|s|Pp+ qx|
∥∥∥
L1([0,x])
≤
√
xe2| Imω|x ‖Pp+ qx‖L2([0,x])
≤
√
xe2| Imω|x ‖v‖1 , (2.25)
one gets that for any x ≥ 0,
|F (x, λ, v)| ≤ e2| Imω|x
(
x
|λ|
e2‖q‖1 +
√
x ‖v‖1
)
. (2.26)
Since the matrix norm is sub-multiplicative, one obtains
|Mn+1(x)| ≤ e| Imω|x
∫
0≤x1≤···≤xn≤x
n∏
k=1
|Eω(−xk)Φλ(xk)Eω(xk)| dxndxn−1 · · · dx1
≤ e| Imω|x
n!
|F (x, λ, v)|n.
Hence by (2.26), the series converges normally as claimed and one has
|M(x, λ, v)| ≤ e| Imω|x exp
(
e2| Imω|x
(
x
|λ|
e2‖q‖1 +
√
x ‖v‖1
))
, x ≥ 0.
Since for any given x ≥ 0, Eω(x) and Φλ(x) are analytic in (λ, v) ∈ C∗ ×H1
c
and continuous in (x, λ, v) ∈ [0,∞)× C∗ ×H1
c so is Mn+1 for any n ≥ 0 by the
definition (2.21) and hence M =
∑
n≥0Mn in view of the normal convergence
of the series. It then follows that M is analytic as a map of v and λ with values
in C([0, 2],C2×2). Finally substituting the series into (2.21) and using that by
the normal convergence of the series, sum and integral commute, one gets (2.20).
Since M(x) and Eω(x) are continuous, (2.12) holds in the L2 sense. It then
follows that M and ∂xM are continuous in v and λ as maps with values in
L2([0, 2],C2×2).
462 Thomas Kappeler and Yannick Widmer
From Theorem 2.2 we derive the following bounds for M = T−1MT .
Corollary 2.3. M is continuous in x, v, λ and for each fixed x, it is analytic
in v and λ. It is the unique solution of (2.3), implying that M and ∂xM are
continuous in v and λ as maps with values in L2([0, 2],C2×2). Furthermore M
satisfies the following estimates for any 0 ≤ x ≤ 2, λ ∈ C∗, v ∈ H1
c ,
|M(x, λ, v)| ≤ e| Imω|x exp
(
e2| Imω|x
(
x
|λ|
e2‖q‖1 +
√
x ‖v‖1
))
and
|M(x,
1
16λ
, v)| ≤ e2‖q‖1+| Imω|x exp
(
e2| Imω|x
(
x
|λ|
e2‖q‖1 +
√
x ‖v‖1
))
.
Proof. Since M = T−1MT , the regularity statements follow from Theorem
2.2. Furthermore, as i√
2
T is unitary one gets
|M(x, λ, v)| ≤ e| Imω|x exp
(
e2| Imω|x
(
x
|λ|
e2‖q‖1 +
√
x ‖v‖1
))
, x ≥ 0.
Since ω( 1
16λ) = ω(−λ) = −ω(λ) and by Proposition 2.1 (ii),
M
(
x,
1
16λ
, q, p
)
= −Re−iRq(x)/2M(x, λ,−q, p)eiRq(0)/2R,
the latter estimate yields∣∣∣∣M (
x,
1
16λ
, q, p
)∣∣∣∣ ≤ e2‖q‖1
∣∣M(x, λ,−q, p)
∣∣
≤ e2‖q‖1+| Imω|x exp
(
e2| Imω|x
(
x
|λ|
e2‖q‖1 +
√
x ‖v‖1
))
.
Next we prove that M is compact in v, uniformly on closed bounded sets of
(x, λ) ∈ [0,∞)× C∗.
Definition 2.4. We call a map from a subset U of a Hilbert space H into
some Banach space compact if it maps sequences in U which converge weakly in
H, to strongly convergent sequences.
Proposition 2.5. For any sequence (vn)n≥1 in H1
c which converges weakly
to an element v∗ in H1
c , one has |M(x, λ, vn) − M(x, λ, v∗)| → 0 as n → ∞,
uniformly on closed bounded subsets of [0,∞)× C∗.
Proof. In view of M = T−1MT , it is enough to prove thatM is compact in v
uniformly on closed bounded sets of (x, λ) ∈ [0,∞)×C∗. In view of the uniform
convergence of the series
∑∞
m=0Mm(x), it suffices to prove the statement for
each term Mm. For M0 = Eω the statement is obviously true, since this term
On Spectral Properties of the L Operator. . . 463
does not depend on v. Now by induction assume that the statement is true for
Mm, and let (vn)n≥0 converge weakly to v∗ in H1
c . By equation (2.21) we have
Mm+1(x, λ, vn) =
∫ x
0
Eω(x− s)Φ(s, λ, vn)Mm(s, λ, vn) ds. (2.27)
By the induction hypothesis |Mm(x, λ, vn) − Mm(x, λ, v∗)| → 0 uniformly on
closed bounded subsets of [0,∞)×C∗. Furthermore, the weak convergence of vn
in H1
c implies that Ppn + (qn)x ⇀ Pp∗ + (q∗)x in L2([0, x]) and qn → q∗, pn →
p∗ in L∞(T). It then follows that cosh(qn) → cosh(q∗), sinh(qn) → sinh(q∗) in
L2([0, x]), yielding that Eω(x−·)Φ(·, λ, vn) weakly converges to Eω(x−·)Φ(·, λ, v∗)
in L2([0, x]), uniformly on bounded subsets of [0,∞) × C∗. Hence the sequence∫ x
0 Eω(x − s)Φ(s, λ, vn)Mm(s, λ, vn)ds converges uniformly on closed bounded
subsets of [0,∞)× C∗ to∫ x
0
Eω(x− s)Φ(s, λ, v∗)Mm(s, λ, v∗) ds =Mm+1(x, λ, v∗).
Since J
(
λ−A−B2/λ
)
is traceless,
detM(x, λ, v) = 1. (2.28)
Hence for any (x, λ, v) ∈ [0,∞] × C∗ × H1
c , M(x, λ, v) =
(
m1 m2
m3 m4
)
is invertible
and M−1(x, λ, v) =
(
m4 −m2
−m3 m1
)
with
|M−1(x, λ, v)| ≤ 2 max(|m4|+ |m2|, |m3|+ |m1|). (2.29)
Proposition 2.6. The λ-derivative Ṁ of M is given by
Ṁ(x) = M(x)
∫ x
0
M−1(s)J(1 +B2/λ2)M(s) ds .
In particular, for any x ≥ 0, Ṁ(x, λ, v) is analytic on C∗×H1
c and on any closed
bounded subset of [0,∞)× C∗ ×H1
c , it is compact and bounded.
Proof. Taking the λ-derivative on both sides of equation (2.3) one sees that
the λ-derivative Ṁ of M fulfills
∂xṀ = J
(
λ−A−B2/λ
)
Ṁ + J(1 +B2/λ2)M
with Ṁ(0, λ, v)(0) = 0. The solution of this inhomogeneous linear equation for
Ṁ is then given by
Ṁ(x) = M(x)
∫ x
0
M−1(s)J(1 +B2/λ2)M(s)ds.
From this formula and the properties established for M , the remaining statements
for Ṁ follow.
464 Thomas Kappeler and Yannick Widmer
Next we establish bounds for the difference of the fundamental solution M
with M(x, λ, 0) = Eω(x) for |λ| large and small. First we need to establish the
following auxiliary result.
Lemma 2.7. For any v ∈ H1
c and any (x, λ) ∈ [0, 1]× C∗,
F (x, λ, v) =
∫ x
0
Eω(−s)Φλ(s)Eω(s) ds
satisfies
|Eω(x)F (x, λ, v)| ≤ 1
|λ|
e| Imω(λ)|xe2‖q‖1 +max
±
∣∣∣∣∫ x
0
(Pp(s) + qx(s)) e±iω(λ)(x−2s)ds
∣∣∣∣.
Proof. Multiply (2.24) by Eω(x) to get
Eω(x)F (x, λ, v) =
∫ x
0
i
16λ
(
e−iωx(cosh(q(s))− 1) − sinh(q(s))e−iω(x−2s)
sinh(q(s))eiω(x−2s) eiωx(1− cosh(q(s)))
)
+
1
4
(Pp(s) + qx(s))
(
e−iω(x−2s)
eiω(x−2s)
)
ds. (2.30)
Hence
|Eω(x)F (x, λ, v)| ≤
∫ x
0
1
|λ|
e| Imω|x (|cosh(q(s))− 1|+ |sinh(q(s))|) dx
+
∣∣∣∣∫ x
0
(Pp(s) + qx(s)) e| Imω|(x−2s)ds
∣∣∣∣.
Using that |cosh(q(s))− 1|+ |sinh(q(s))| ≤ e2‖q‖1 yields the claim.
For (x, λ, v) ∈ [0, 1]× C∗ ×H1
c , let
M̂(x, λ, v) :=M(x, λ, v)− Eω(x).
Lemma 2.8. On [0, 1]×H1
c for all λ ∈ C with |λ| ≥ 1/4,
|M̂(x, λ, v)| ≤ |Eω(x)F (x, λ, v)|
+ Cve
| Imω|x
(∫ x
0
e−2| Imω|s|Eω(s)F (s, λ, v)|2 ds
)1/2
,
where Cv = cec and c = ‖v‖1 + e2‖q‖1 .
Proof. By (2.20),
M̂(x, λ, v) =
∫ x
0
Eω(x− s)Φλ(s)M(s) ds
satisfies
M̂(x, λ, v) =
∫ x
0
Eω(x− s)Φλ(s)Eω(s) ds+
∫ x
0
Eω(x− s)Φλ(s)M̂(s) ds,
On Spectral Properties of the L Operator. . . 465
yielding
M̂(x, λ, v) = Eω(x)F (x, λ, v) +
∫ x
0
Eω(x− s)Φλ(s)M̂(s)ds. (2.31)
Clearly, ∣∣Eω(x− s)Φλ(s)M̂(s)
∣∣ ≤ e| Imω|(x−s)∣∣Φλ(s)
∣∣∣∣M̂(s)
∣∣.
It is convenient to introduce the following weighted norm for a x-dependent 2× 2
matrix ∣∣A(x)
∣∣
ω
:= e−| Imω|x∣∣A(x)
∣∣.
Multiplying both sides of (2.31) by e−| Imω|x one obtains the following estimate
∣∣M̂(x)
∣∣
ω
≤
∣∣Eω(x)F (x, λ, v)
∣∣
ω
+
∫ x
0
∣∣Φλ(s)
∣∣∣∣M̂(s)
∣∣
ω
ds
and hence by Gronwall’s inequality and the estimate
|Φ(s)| ≤
(
|Pp(s) + qx(s)|+ 1
|λ|
e|q(s)|
)
=: b(s, λ)
we get ∣∣M̂(x)
∣∣
ω
≤
∣∣Eω(x)F (x, λ, v)
∣∣
ω
+
∫ x
0
∣∣Eω(s)F (s, λ, v)
∣∣
ω
b(s, λ) exp
(∫ x
s
b(r, λ)dr
)
ds.
Arguing as in (2.25) with x = 1, one gets∫ 1
0
b(r, λ)dr ≤ ‖v‖1 +
1
|λ|
e2‖q‖1 .
An application of the Cauchy–Schwarz inequality then yields the claim.
We now use Lemma 2.8 to derive estimates for M and Ṁ from those of F .
Theorem 2.9. M and Ṁ have the following asymptotics:
(i) For |λ| → ∞, locally uniformly on [0, 1]×H1
c ,
M(x, λ, v) = Eω(λ)(x) + o(e| Imω(λ)|x),
Ṁ(x, λ, v) = Ėω(λ)(x) + o(e| Imω(λ)|x),
where Ėω(λ)(x) = −iω̇(λ)REω(λ)(x) and ω̇(λ) = 1 + 1
16λ2 .
(ii) For |λ| → ∞, uniformly on [0, 1] and on bounded subsets of H2
c ,
M(x, λ, v) = Eω(λ)(x) +O(e| Imω(λ)|x/|ω(λ)|),
Ṁ(x, λ, v) = Ėω(λ)(x) +O(e| Imω(λ)|x/|ω(λ)|).
466 Thomas Kappeler and Yannick Widmer
Proof. (i) In view of Lemma 2.8 it remains to prove an appropriate asymptotic
estimate for |Eλ(x)F (x, λ, v)|. By Lemma 2.7
|Eω(λ)(x)F (x, λ, v)| ≤ 1
|λ|
e| Imω(λ)|xe2‖q‖1
+ max
±
∣∣∣∣∫ x
0
(Pp(s) + qx(s)) e±iω(λ)(x−2s) ds
∣∣∣∣. (2.32)
For arbitrary ε > 0 there is ωε > 0, depending locally uniformly on v ∈ H1
c , such
that for any λ ∈ C∗ with |ω(λ)| > ωε and 0 ≤ x ≤ 1 one has (cf. Lemma D.1
in [3]) ∣∣∣∣∫ x
0
(
Pp(s) + qx(s)
)
e±iω(x−2s) ds
∣∣∣∣ ≤ εe| Imω|x,
yielding the stated asymptotics ofM. The claimed asymptotics for Ṁ is obtained
by applying Cauchy’s estimate to the λ-derivative of M̂.
(ii) In case v ∈ H2
c , ∫ x
0
(
Pp(s) + qx(s)
)
e±iω(x−2s) ds
can be integrated by parts. Using that for ω ≡ ω(λ) 6= 0,
e±iω(x−2s) =
−1
±2iω
d
ds
e±iω(x−2s)
one gets∫ x
0
(
Pp(s) + qx(s)
)
e±iω(x−2s)ds = − 1
±2iω
(
(Pp(x) + qx(x))e∓iωx
+ (Pp(0) + qx(0))e±iωx
)
−
∫ x
0
(Ppx(s) + qxx(s))e±iω(x−2s) ds.
Since by (2.23), ‖qx‖L∞ ≤ 2 ‖q‖2 and
‖Pp‖L∞ ≤ 2 ‖Pp‖1 = 2
(∑
k
〈2k〉2|〈2k〉p̂2k|2
)1/2
= 2 ‖p‖2
one gets for any 0 ≤ x ≤ 1, |ω(λ)| > 0 and v ∈ H2
c
|Eω(λ)(x)F (x, λ, v)| ≤ 1
|λ|
e| Imω(λ)|xe2‖q‖1
+
1
2|ω(λ)|
(
2 ‖v‖2 e
| Imω(λ)|x + 2 ‖v‖2 e
| Imω(λ)|x
)
.
The claimed asymptotics for Ṁ is once more obtained by applying Cauchy’s
estimate to the λ-derivative of M̂.
Theorem 2.10. For bi-infinite sequences of complex numbers (ζn)n ⊂ C∗
with |ζn| ≥ 1
4 , the following holds:
On Spectral Properties of the L Operator. . . 467
(i) If ζn = nπ +O(1), then
M(x, ζn, v) = Eω(ζn)(x) + `2n and Ṁ(x, ζn, v) = Ėω(ζn)(x) + `2n,
where Eω(ζn)(x) = e−Rω(ζn)x and ω̇(ζn) = 1 + 1
16ζ2
n
, implying that
Ėω(ζn)(x) = −xω̇(ζn)Re−Rω(ζn)x = −xREω(ζn)(x) + `1n.
These estimates hold uniformly on [0, 1], on bounded subsets of H1
c and on
subsets of sequences (ζn)n where (ω(ζn) − nπ)n is bounded in `∞C . In more
detail, e.g., the first estimate means that for any bounded subset V ⊂ H1
c and
any subset B of sequences (ζn)n ⊂ C∗, with (ω(ζn) − nπ)n bounded in `∞C ,
there exists C > 0 so that
sup
0≤x≤1
∑
n∈Z
∣∣M(x, ζn, v)− Eω(ζn)(x)
∣∣2 ≤ C, for any v ∈ V and (ζn)n ∈ B.
(ii) If ζn = nπ + `2n, then
M(x, ζn, v) = Enπ(x) + `2n and Ṁ(x, ζn, v) = −xREnπ(x) + `2n.
These estimates hold uniformly on [0, 1], on bounded subsets of H1
c , and on
subsets of sequences (ζn)n where (ω(ζn)− nπ)n is bounded in `2C.
Proof. (i) By Lemma 2.8, on [0, 1]×H1
c for any |λ| ≥ 1/4
|M(x, λ, v)− Eω(x)| ≤ O
(
e| Imω|‖Eω(·)F (·, λ, v)‖L∞([0,1])
)
.
By Lemma 2.7,
|Eω(ζn)(x)F (x, ζn, v)| ≤ 1
|ζn|
e| Imω(ζn)|xe2‖q‖1
+ max
±
∣∣∣∣∫ x
0
(
Pp(s) + qx(s)
)
e±iω(ζn)(x−2s) ds
∣∣∣∣ ,
and by Lemma D.2 in [3],
∑
n∈Z
max
±
∣∣∣∣∫ x
0
(
Pp(s) + qx(s)
)
e±iω(ζn)(x−2s) ds
∣∣∣∣2 ≤ 2e2b ‖v‖1 , 0 ≤ x ≤ 1,
where b = supn∈Z |ω(ζn) − nπ|. Altogether, we thus have proved∑
n∈Z |M(x, ζn, v)− Eω(ζn)(x)|2 <∞. In view of Lemma 2.8, the latter estimate
holds uniformly on bounded subsets of H1
c and on subsets of sequences (ζn)n in
C∗ so that (ω(ζn) − nπ)n is bounded in `∞C . To obtain the claimed estimate for
Ṁ(x, ζn, v)−Ėω(ζn)(x) we apply Cauchy’s estimate to λ-discs Dζn of fixed radius
around each ζn to get
∣∣Ṁ(x, ζn, v)− Ėω(ζn)(x)
∣∣ ≤ O( sup
λ∈Dζn
|M(x, λ, v)− Eω(λ)(x)|
)
.
468 Thomas Kappeler and Yannick Widmer
Since the radii of the discs Dζn do not depend on n one then concludes that
sup
0≤x≤1
∑
n∈Z
∣∣Ṁ(x, ζn, v)− Ėω(ζn)(x)
∣∣2 <∞,
where the estimate holds uniformly in the claimed sense. Altogether this estab-
lishes the first claim.
(ii) For sequences (ζn)n with the stronger asymptotics ω(ζn) = nπ + `2n, we
have
Eω(ζn)(x) = e−Rω(ζn)x = e−Rnπx + `2n = Eω(nπ)(x) + `2n
implying the claimed estimate.
Theorem 2.11. Uniformly on bounded subsets of H2
c and on subsets of bi-
infinite sequences of complex numbers with ζn = nπ +O(1), |ζn| > 1
4 ,
sup
0≤x≤1
∣∣M(x, ζn, v)− Eω(ζn)(x)
∣∣ = O (1/n) ,
sup
0≤x≤1
∣∣Ṁ(x, ζn, v)− Ėω(ζn)(x)
∣∣ = O (1/n) .
If in addition ζn = nπ + `2n, then
sup
0≤x≤1
∣∣M(x, ζn, v)− Enπ(x)
∣∣ = `2n,
sup
0≤x≤1
∣∣Ṁ(x, ζn, v)− Ėnπ(x)
∣∣ = `2n.
Note that Ėω(ζn) = −xω̇(ζn)Re−Rω(ζn)x = −xREω(ζn)(x) +O
(
1
n
)
.
Proof. The first estimate is a consequence of Theorem 2.9 and the second one
is then obtained by applying Cauchy’s estimate. If the sequence ζn satisfies in
addition ζn = nπ + `2n then Eω(ζn)(x) = Enπ(x) + `2n.
For later reference we state the asymptotics of M = T−1MT , corresponding
to the ones obtained for M. Introduce
Eω(λ)(x) := M(x, λ, 0) = T−1Eω(λ)(x)T =
(
cos(ω(λ)x) sin(ω(λ)x)
− sin(ω(λ)x) cos(ω(λ)x)
)
. (2.33)
Since
Ėω(λ)(x) = −T−1xω̇(λ)Re−Rω(λ)xT = −xω̇(λ)T−1RTEω(λ)(x)
= xJEω(λ)(x) +O
(
e| Imω(λ)|x/|λ|2
)
. (2.34)
Theorems 2.9–2.11 then yield the following results.
Theorem 2.12. M and Ṁ have the following asymptotics:
On Spectral Properties of the L Operator. . . 469
(i) For |λ| → ∞ locally uniformly on [0, 1]×H1
c ,
M(x, λ, v) = Eω(λ)(x) + o
(
e| Imω(λ)|x
)
,
Ṁ(x, λ, v) = xJEω(λ)(x) + o
(
e| Imω(λ)|x
)
.
(ii) For |λ| → ∞ uniformly on bounded subsets of [0, 1]×H2
c ,
M(x, λ, v) = Eω(λ)(x) +O
(
e| Imω(λ)|x/|ω(λ)|
)
,
Ṁ(x, λ, v) = xJEω(λ)(x) +O
(
e| Imω(λ)|x/|ω(λ)|
)
.
Furthermore for bi-infinite sequences of complex numbers (ζn)n ⊂ C∗ with |ζn| ≥
1
4 one has:
(iii)If ζn = nπ +O(1), then
M(x, ζn, v) = Eω(ζn)(x) + `2n and Ṁ(x, ζn, v) = xJEω(ζn)(x) + `2n,
uniformly on [0, 1], on bounded subsets of H2
c , and on subsets of sequences
(ζn)n where (ω(ζn)− nπ)n is bounded in `∞C .
(iv) If ζn = nπ + `2n, then
M(x, ζn, v) = Enπ(x) + `2n and Ṁ(x, ζn, v) = xJEnπ(x) + `2n,
uniformly on [0, 1], on bounded subsets of H1
c , and on subsets of sequences
(ζn)n where (ω(ζn)− nπ)n is bounded in `2C.
(v) If ζn = nπ +O(1) and v ∈ H2
c , then
M(x, ζn, v) = Eω(ζn)(x) +O (1/n) and Ṁ(x, ζn, v) = xJEω(ζn)(x) +O (1/n)
uniformly on [0, 1], on bounded subsets of H2
c , and on subsets of sequences ζn
where (ζn − nπ)n is bounded in `∞C .
Recall that by (2.29), M−1 =
(
m4 −m2
−m3 m1
)
. Furthermore, E−1
ω(λ) = E−ω(λ) =
Eω(−λ) and hence
(E−1
ω(λ))
· = −Ėω(−λ)(x) = −xJEω(−λ)(x) +O(e| Imω(−λ)|x/λ2).
Theorem 2.12 then leads to the following results for M−1.
Corollary 2.13. M−1 and (M−1)· satisfy the following estimates:
(i) For |λ| → ∞ locally uniformly on [0, 1]×H1
c ,
M−1(x, λ, v) = Eω(−λ)(x) + o
(
e| Imω(λ)|x
)
,
(M−1)·(x, λ, v) = −xJEω(−λ)(x) + o
(
e| Imω(λ)|x
)
.
470 Thomas Kappeler and Yannick Widmer
(ii) For |λ| → ∞,
M−1(x, λ, v) = Eω(−λ)(x) + o
(
e| Imω(λ)|x/|ω(λ)|
)
uniformly on bounded subsets of [0, 1]×H2
c , and
(M−1)·(x, λ, v) = −xJEω(−λ)(x) + o
(
e| Imω(λ)|x/|ω(λ)|
)
.
Furthermore for bi-infinite sequences of complex numbers (ζn)n in C∗ with
|ζn| ≥ 1
4 one has:
(iii)If ζn = nπ +O(1), then
M−1(x, ζn, v) = Eω(−ζn)(x) + `2n and (M−1)·(x, ζn, v) = −xJEω(−ζn)(x) + `2n
uniformly on [0, 1], on bounded subsets of H1
c , and on subsets of sequences
(ζn)n, where (ω(ζn)− nπ)n is bounded in `∞C .
(iv) If ζn = nπ + `2n, then
M−1(x, ζn, v) = E−nπ(x) + `2n and (M−1)·(x, ζn, v) = −xJE−nπ(x) + `2n
uniformly on [0, 1], on bounded subsets of H1
c , and on subsets of sequences
(ζn)n, where (ω(ζn)− nπ)n is bounded in `2C.
(v) If ζn = nπ +O(1) and v ∈ H2
c , then
M−1(x, ζn, v) = Eω(−ζn)(x) +O(1/n)
and
(M−1)·(x, ζn, v) = −xJEω(−ζn)(x) +O(1/n)
uniformly on [0, 1], on bounded subsets of H2
c , and sequences (ζn)n, where
(ζn − nπ)n is bounded in `∞C .
In order to study the periodic spectrum of the operator Q, its discriminant
plays an important role. For any v ∈ H1
c , let
M̀(λ, v) := M(x, λ, v)|x=1 and M̀(λ, v) =:
(
m̀1 m̀2
m̀3 m̀4
)
(2.35)
as well as M̀(λ, v) := M(1, λ, v). The discriminant and anti-discriminant are
then defined as follows:
∆(λ, v) :=
1
2
trM̀(λ, v) =
1
2
trM̀(λ, v), δ(λ, v) := (m̀1(λ, v)−m̀4(λ, v))/2. (2.36)
Lemma 2.14. ∆ and δ are analytic maps on C∗×H1
c and have the following
symmetries: for any λ ∈ C∗ and v ∈ H1
c ,
On Spectral Properties of the L Operator. . . 471
(i) (Reflection in λ)
∆(−λ, v) = ∆(λ, v), δ(−λ, v) = δ(λ, v).
(ii) (Reciprocity in λ)
∆(
1
16λ
, q, p) = ∆(λ,−q, p), δ(
1
16λ
, q, p) = δ(λ,−q, p).
(iii)(Conjugation)
∆(λ, v) = ∆(λ, v), δ(λ, v) = δ(λ, v).
(iv) (Reflection of v)
∆(λ,−v) = ∆(λ, v), δ(λ,−v) = −δ(λ, v).
(v) (Real potentials) If the components q and p of v are real valued then ∆(λ, v)
and δ(λ, v) are real for any λ ∈ R ∪ iR.
(vi) (Purely imaginary potentials) If q and p take values in iR then ∆(λ, v) is real
and δ(λ, v) is purely imaginary for any λ ∈ R ∪ iR.
Proof. Items (i)–(iv) follow from Proposition 2.1.
(v) By (i) and (iii) one has for v real that for any λ ∈ R ∪ iR
∆(λ, v) = ∆(λ, v) = ∆(λ, v) and δ(λ, v) = δ(λ, v) = δ(λ, v) .
(vi) In case v is purely imaginary it follows from (i), (iii), and (iv) that for
any λ ∈ R ∪ iR
∆(λ, v) = ∆(λ,−v) = ∆(λ, v) and δ(λ, v) = δ(λ,−v) = −δ(λ, v).
The latter lemma leads to the following
Corollary 2.15. Discriminant and anti-discriminant together with their λ-
derivatives are real analytic on C∗ ×H1
c . On any closed, bounded subset of C∗ ×
H1
c , ∆ and δ are compact and bounded. More precisely, for any compact subset
K ⊂ C∗ and any closed, bounded subset V ⊂ H1
c , the map V → L∞C (K), v 7→
(λ 7→ ∆(λ, v)) is compact in the sense of Definition 2.4.
For later reference we record the following formulas for ∆(λ, v) and δ(λ, v) at
the zero potential v = 0. Recall that ω(λ) = λ − 1
16λ . Taking into account that
by (2.33), M̀(λ, 0) = Eω(λ)(1), the following holds.
Lemma 2.16. For any λ ∈ C∗
∆(λ, 0) = cos(ω(λ)), ∆̇(λ, 0) = −
(
1 +
1
16λ2
)
sin(ω(λ)), δ(λ, 0) = δ̇(λ, 0) = 0.
As a consequence ∆2(λ, 0)− 1 = − sin2(ω(λ)).
472 Thomas Kappeler and Yannick Widmer
To obtain rough asymptotics of the periodic eigenvalues we need to compare
∆(λ, v) with ∆(λ, 0). Recall that the domains Dn, n ∈ Z, were introduced in the
introduction (cf. also (3.1) below).
Lemma 2.17. For any given v ∈ H1
c , the following asymptotics for |λ| → ∞
hold on C \
⋃
n≥1Dn ∪ (−Dn)
∆2(λ)− 1 = − sin2(ω(λ)) (1 + o(1)) = − sin2(λ) (1 + o(1)) , (2.37)
∆̇(λ) = − sin(λ) (1 + o(1)) . (2.38)
These estimates hold locally uniformly on H1
c .
Proof. By Theorem 2.12 (ii), ∆(λ, v) = cos(ω(λ)) + o
(
e| Imω(λ)|) , and thus
∆2(λ, v)− 1
= − sin2(ω(λ))
(
1 +
o
(
e| Imω(λ)|) cos(ω(λ))
sin2(ω(λ))
+
o
(
e2| Imω(λ)|)
sin2(ω(λ))
)
. (2.39)
For λ ∈ C∗ \
⋃
n≥1Dn ∪ (−Dn) there exists m ∈ Z with mπ+π/3 ≤ Reλ ≤ (m+
1)π − π/3. If in addition, λ is sufficiently large, then ω(λ) = λ − 1
16λ satisfies
mπ + π/4 ≤ Reω(λ) ≤ (m+ 1)π − π/4. Hence | sin(Reω(λ))| ≥ 1√
2
and
| sin(ω(λ))| = | sin(Reω(λ)) cos(i Imω(λ)) + cos(Reω(λ) sin(i Imω(λ))|
≥ cosh(| Imω(λ)|)/
√
2 ≥ e| Imω(λ)|/
√
2.
It follows that for λ ∈ C∗ \
⋃
n≥1Dn ∪ (−Dn) sufficiently large∣∣∣∣cos(ω(λ))
sin(ω(λ))
∣∣∣∣ ≤ e| Imω(λ)|
| sin(ω(λ))|
≤
√
2,
and hence the expression inside the large parentheses of (2.39) is 1 + o(1). The
asymptotics (2.37) then follow since
sin(ω(λ)) = sin(λ) cos
(
1
16λ
)
− sin
(
1
16λ
)
cos(λ)
= sin(λ)
(
1 +O
(
1
λ2
)
+O
(
1
λ
)
cos(λ)
sin(λ)
)
and ∣∣∣∣cos(λ)
sin(λ)
∣∣∣∣ ≤ e| Imλ|
1√
2
e| Imλ| ≤
√
2.
Concerning (2.38), note that by Theorem 2.12 (i),
Ṁ(x, λ, v) = xJEω(λ)(x) + o
(
e| Imω(λ)|x
)
implying that
∆̇(λ, v) = − sin(ω(λ)) + o
(
e| Imω(λ)|
)
.
A similar argument as the one above then yields the claimed asymptotics.
On Spectral Properties of the L Operator. . . 473
3. Spectra
The main purpose of this section is to study the asymptotics of the pe-
riodic, the Dirichlet, and the Neumann eigenvalues of the operator Q(v) for
v ∈ H1
c . They provide the setup for the proofs of Theorem 1.1 and Theo-
rem 1.2 in Section 4. Recall that in Section 1, we introduced the domains D0 :={
z ∈ C : |z − 1
4 | <
1
4π
}
and for any n ≥ 1,
Dn := {λ ∈ C : |λ− nπ| < π/3} , D−n :=
{
λ ∈ C :
1
16λ
∈ Dn
}
(3.1)
as well as B0 := {λ ∈ C : |λ| ≤ π/2} and for any n ≥ 1 and
Bn := {λ ∈ C : |λ| < nπ + π/2} ,
B−n :=
{
λ ∈ C : |λ| ≤ 1
16(nπ + π/2)
}
, (3.2)
and we denote by An the open annulus
An := Bn \B−n. (3.3)
Furthermore, by the definition (2.35)
M̀(λ, v) = M(x, λ, v)|x=1 and M̀(λ, v) =
(
m̀1 m̀2
m̀3 m̀4
)
(3.4)
as well as M̀(λ, v) = M(x, λ, v)|x=1. Denote by Qdir the operator Q = Q1∂x +
Q0 with domain
Hdir :=
{
F = (F1, F2, F3, F4) ∈ H1([0, 1],C4) : F1(0) = F1(1) = 0
}
.
Its spectrum is discrete and coincides with the Dirichlet spectrum of the spectral
problem (2.2), defined as the set of eigenvalues with eigenfunctions f = (f1, f2) ∈
H1([0, 1],C2) such that f1(0) = 0 = f1(1). Clearly, for any v ∈ H1
c , µ ∈ C∗ is a
Dirichlet eigenvalue of (2.2) if there exists a ∈ C∗ such that
M̀(µ, v)
(
0
1
)
= a
(
0
1
)
. (3.5)
We thus have the following
Theorem 3.1. The Dirichlet spectrum of Q(v) with v ∈ H1
c is the zero
set of the function χD(λ) := m̀2(λ), {µ ∈ C∗ : χD(µ) = 0}. Furthermore, the
multiplicity Mult(µ, χD) of a root µ of m̀2 is equal to the algebraic multiplicity
Multa(µ) of µ as a Dirichlet eigenvalue, defined as the dimension of the (finite
dimensional) vector space
⋃
n≥1 ker(µ−Qdir(v))n. The function χD is an analytic
and compact function on C∗ ×H1
c . For v = 0, χD(λ, 0) = sin(ω(λ)).
474 Thomas Kappeler and Yannick Widmer
All the assertions of Theorem 3.1 are shown in a straightforward way except
the one on the multiplicity of the roots of χD, which we state separately in
Lemma 3.5 below. To prove it, we first need to discuss some elementary properties
of the Dirichlet eigenvalues and χD.
Lemma 3.2. For any (λ, v) ∈ C∗ ×H1
c
(i) χD(−λ, v) = −χD(λ, v) and χD( 1
16λ , q, p) = −e−q(0)χD(λ,−q, p).
(ii) χD(λ, v) = χD(λ, v) and χD(λ,−v) = −m̀3(λ, v).
(iii)For |λ| → ∞ with λ 6∈
⋃
n≥1Dn∪(−Dn), χD(λ, v) = χD(λ, 0)(1+o(1)) locally
uniformly in v ∈ H1
c .
Proof. (i) Using that(
χD(λ, v)
0
)
=
(
m̀2(λ, v)
0
)
=
(
1 0
0 0
)
M̀(λ, v)
(
0
1
)
,
we obtain by Proposition 2.1(
χD(−λ, v)
0
)
=
(
1 0
0 0
)
M̀(−λ, v)
(
0
1
)
= −
(
1 0
0 0
)
RM̀(λ, v)R
(
0
1
)
=
(
−m̀2(λ, v)
0
)
,(
χD( 1
16λ , v)
0
)
=
(
1 0
0 0
)
M̀(
1
16λ
, q, p)
(
0
1
)
= −
(
1 0
0 0
)
ReiRq(0)/2M̀(λ,−q, p)e−iRq(0)/2R
(
0
1
)
= −
(
ie−q(0)/2 0
0 0
)
M̀(λ,−q, p)
(
0
−ie−q(0)/2
)
= −
(
e−q(0)m̀2(λ,−q, p)
0
)
.
(ii) is proved in a similar way as item (i).
(iii) By the same argument as in the proof of Lemma 2.17 one obtains the
claimed asymptotics χD(λ, v) = χD(λ, 0)(1 + o(1)).
As usual we denote by +
√
λ the principal branch of the square root defined for
λ in C \ (−∞, 0] and determined by +
√
1 = 1.
Lemma 3.3. The Dirichlet eigenvalues at v = 0 are 1
4 ,−
1
4 and
nπ
2
(
+
√
1 +
1
4n2π2
+ 1
)
,
nπ
2
(
+
√
1 +
1
4n2π2
− 1
)
, n 6= 0,
each eigenvalue having multiplicity one.
On Spectral Properties of the L Operator. . . 475
A rough localization of the Dirichlet eigenvalues is provided by the following
Lemma 3.4 (Counting Lemma). For each potential in H1
c , there exist a
neighborhood U in H1
c and an integer N > 0 so that for any v ∈ U , the function
λ 7→ χD(λ, v) has exactly one root in each of the domains Dn,−Dn, D−n,−D−n
for any n > N and exactly 2 + 4N in the annulus AN , counted with their multi-
plicities. There are no other roots.
Proof. By Lemma 3.2, for |λ| → ∞ with λ 6∈
⋃
n≥1Dn ∪ (−Dn), χD(λ, v) =
χD(λ, 0) (1 + o(1)) locally uniformly in v. Hence, for any potential in H1
c there
is a neighborhood U and an integer N ≥ 1 such that for any v ∈ U
|χD(λ, v)− χD(λ, 0)| < |χD(λ, 0)|, (3.6)
|χD(λ, (−q, p))− χD(λ, 0)| < |χD(λ, 0)| (3.7)
on the boundaries of the discs Dn,−Dn, and Bn for any n ≥ N . It follows by
Rouché’s theorem that χD(·, v) has as many roots inside any of the discs ±Dn,
n ≥ N , as χD(·, 0). There are no other roots in C∗ \
(
BN
⋃
n≥N (Dn ∪ −Dn)
)
.
By Lemma 3.2 (i)∣∣∣∣eq(0)χD(
1
16λ
, (q, p))− χD(
1
16λ
, 0)
∣∣∣∣ = |χD(λ, (−q, p))− χD(λ, 0)|
< |χD(λ, 0)|. (3.8)
Since χD(·, v) has the same roots as eq(0)χD(·, v) and χD( 1
16λ , 0) = χD(λ, 0)
it follows that χD(λ, v) has as many roots as χD(λ, 0) inside any of the discs
D−n,−D−n with n > N . It remains to count the roots inside An with n ≥ N .
In order to apply Rouché’s theorem we need to estimate χD on the boundary of
B−n. Arguing as above one concludes that for any λ on the boundary of Bn with
n ≥ N and t, t+ s ∈ [0, 1],∣∣∣∣e(t+s)q(0)χD
(
1
16λ
, (t+ s)v
)
− etq(0)χD
(
1
16λ
, tv
)∣∣∣∣
= |χD(λ,−(t+ s)v)− χD(λ,−tv)|
<
1
2
|χD(λ,−tv)| = 1
2
∣∣∣∣etq(0)χD
(
1
16λ
, tv
)∣∣∣∣ .
After division by |etq(0)| one gets∣∣∣∣esq(0)χD
(
1
16λ
, (t+ s)v
)
− χD
(
1
16λ
, tv
)∣∣∣∣ ≤ 1
2
∣∣∣∣χD ( 1
16λ
, tv
)∣∣∣∣ .
Choose ε > 0 such that∣∣∣∣esq(0) − 1||χD
(
1
16λ
, (t+ s)v
)∣∣∣∣ < 1
2
∣∣∣∣χD ( 1
16λ
, tv
)∣∣∣∣
476 Thomas Kappeler and Yannick Widmer
for λ on the boundary of Bn with n ≥ N , t ∈ [0, 1], and 0 ≤ s < ε. Then
|χD(λ, (t+ s)v)− χD(λ, tv)| < |χD(λ, tv)|
on the boundary of An. By Rouché’s Theorem it then follows that the number
of roots of χD(·, tv) inside An is independent of t ∈ [0, 1]. Since χD(·, 0) has 2 +
4N roots inside AN so does χD(·, v). Since (An)n≥N is a covering of C∗, there
are no roots in C∗ \
(
AN ∪
⋃
n≥N Dn ∪ (−Dn)
)
.
Since by Lemma 3.2 (i), χD(−λ, v) = −χD(λ, v), it is enough to consider the
Dirichlet eigenvalues of Q(v) in
C+ := {λ ∈ C : Reλ > 0} ∪ iR>0. (3.9)
For any v ∈ H1
c , these eigenvalues, when counted with their multiplicities
Multa(µ), can be listed as a bi-infinite sequence
0 � · · · � µ−2 � µ−1 � µ0 � µ1 � µ2 � · · · .
Here � is the ordering of complex numbers in C+defined as follows: for a, b ∈
C+, a � b, [
|a| < |b|
]
or
[
|a| = |b| and Im a ≤ Im b
]
. (3.10)
Note that � is a total ordering of C+. One of its feature is that for any a ∈ C+,
a � i|a|. In particular, ordering the Dirichlet eigenvalues in this way one has that
µn = nπ + o(1) and 1
16µ−n
= nπ + o(1).
Lemma 3.5. For any Dirichlet eigenvalue µ of Q(v) with v ∈ H1
c ,
Multa(µ) = Mult(µ, χD).
Proof. The algebraic multiplicity Multa(µ) of µ equals the dimension of the
range of the Riesz projector
Π(µ) :=
1
2πi
∫
Γ(µ)
(λ−Qdir(v))−1 dλ,
where Γ(µ) is a counterclockwise oriented contour around µ so that all Dirich-
let eigenvalues of Q(v) except µ are outside of Γ(µ). Since (λ − Qdir(v))−1 is
a compact operator, Multa(µ) is finite and Multa(µ) = tr Π(µ). By Propo-
sition 2.1 and Lemma 3.2, Multa(µn) = Multa(−µn) and Mult(µn, χD) =
Mult(−µn, χD). Hence it suffices to consider the Dirichlet eigenvalues in C+.
By Lemma 3.3, the Dirichlet eigenvalues for v = 0 contained in C+ are given
by µ0
k = 1
2
(
kπ +
√
k2π2 + 1/4
)
, k ∈ Z, and since χD(λ, 0) = sin(ω(λ)) one has
Mult(µ0
k, χD) = 1. Note that for any k ∈ Z,(
sin(ω0
kx), cos(ω0
kx),
1
4µ0
k
sin(ω0
kx),
1
4µ0
k
cos(ω0
kx)
)
is an eigenfunction of Qdir(0), corresponding to the eigenvalue µ0
k, where ω0
k ≡
ω(µ0
k) = π|k|. Since Qdir(0) is selfadjoint with respect to the canonical inner
On Spectral Properties of the L Operator. . . 477
product on L2
(
[0, 1],C4
)
, the algebraic multiplicity Multa(µ
0
k) is one. Since
Mult(µ0
k, χD) = 1 it then follows that Multa(µ
0
k) = Mult(µ0
k, χD) for any k ∈
Z. Now let v0 ∈ H1
c and consider the line segment [0, v0] from 0 to v0 in H1
c .
Since it is compact it follows by the Counting Lemma that there exist a neigh-
borhood U of [0, v0] in H1
c and N ≥ 1 such that for any potential v in U and
|k| > N , µk(v) ∈ Dk. It implies that Mult(µk, χD) = 1. Choosing Γ(µn) := ∂Dn
as contour for the Riesz projector introduced above, one sees that Multa(µk) = 1
for any |k| > N . For the remaining 4N + 2 Dirichlet eigenvalues in AN consider
the Riesz projector
ΠN (v) :=
1
2πi
∫
∂AN
(λ−Qdir(v))−1 dλ.
Denote by RN (v) the range of ΠN (v) and let ΛN (v) = Q(v)|RN . Since trΠN is
continuous and hence constant in U , the dimension of RN (v) is 4N + 2 and ΛN
maps RN onto itself. Thus, ξN (λ, v) := det (λ− ΛN (v)) is a polynomial of degree
4N + 2. By construction, its roots are precisely the Dirichlet eigenvalues inside
AN , counted with their algebraic multiplicities. On the other hand, consider the
polynomial
ζN (λ, v) :=
∏
|k|≤N
(λ− µk(v))(λ+ µk(v),
formed by the roots µk(v),−µk(v), |k| ≤ N , of χD inside AN counted with their
multiplicities Mult(µk, χD). By the analyticity of χD and the argument principle,
the coefficients of ζN are in fact analytic functions in v ∈ U . The same is true
for the coefficients of ξN . By the same argument as in Lemma 3.4 there is a
neighborhood U (0) of 0 in H1
c so that on U (0), µk ∈ Dk for any k ∈ Z. Hence ξN
and ζN coincide on U (0) ∩ U (6= ∅). By the analyticity of the coefficients of ζN
and ξN we conclude that ξN (·, v) = ζN (·, v) for all v ∈ U (0) ∩ U , implying that
on U , Multa(µk(v)) = Mult(µk(v), χD) for any |k| ≤ N .
Denote by Qneu the operator Q(v) with domain
Hneu :=
{
F = (F1, F2, F3, F4) ∈ H1
(
[0, 1],C4
)
: F2(0) = 0 = F2(1)
}
.
Its spectrum, referred to as Neumann spectrum, is discrete and coincides with the
Neumann spectrum of the spectral problem (2.2), defined as the set of eigenvalues
with eigenfunctions f = (f1, f2) ∈ H1([0, 1],C2) such that f2(0) = 0 = f2(1).
Clearly, for any v ∈ H1
c , ν ∈ C∗ is an eigenvalue of (2.2) if there exists a ∈ C∗
such that
M̀(λ, v)
(
1
0
)
= a
(
1
0
)
.
Since by Lemma 3.2 (ii), χN (λ, v) = −χD(λ,−v), Theorem 3.1 and Lemma 3.4
yield the following results.
Theorem 3.6. The Neumann spectrum of Q(v) with v ∈ H1
c is the zero
set of the function χN (λ) := m̀3(λ), {ν ∈ C∗ : χN (ν) = 0}. Furthermore, the
478 Thomas Kappeler and Yannick Widmer
multiplicity Mult(ν, χN ) of the root ν equals the algebraic multiplicity Multa(ν)
of ν as a Neumann eigenvalue, i.e., the dimension of the (finite dimensional)
vector space
⋃
n≥1 ker(ν − Qneu(v))n. The function χN is antisymmetric in λ
and hence the Neumann spectrum is even in λ. The function χN is analytic
and compact on C∗ × H1
c . For v = 0, χN (λ, 0) = − sin(ω(λ)). Finally, results
corresponding to Lemma 3.4 also hold for the Neumann eigenvalues.
Lemma 3.3 and Lemma 3.2 (ii) lead to the following
Lemma 3.7. The Neumann spectrum of Q(v) at v = 0 coincides with the
Dirichlet spectrum of Q(v) at v = 0.
The Neumann eigenvalues of Q(v) for v ∈ H1
c , contained in C+and counted
with their algebraic multiplicities can be listed as a bi-infinite sequence
0 � · · · � ν−2 � ν−1 � ν0 � ν1 � ν2 � · · · (3.11)
so that for |k| sufficiently large, νk is the unique Neumann eigenvalue of Q(v) in
the disc Dk.
We finish this section with the following useful identity.
Lemma 3.8. For any Dirichlet or Neumann eigenvalue λ of Q(v) with v ∈
H1
c ,
∆2(λ, v)− 1 = δ2(λ, v).
Proof. By the Wronskian identity 1 = m̀1m̀4 − m̀2m̀3. Hence
∆2 − 1 =
1
4
(m̀1 + m̀4)2 − 1
satisfies
∆2− 1 =
1
4
(m̀1 + m̀4)2 − m̀1m̀4 + m̀2m̀3 =
1
4
(m̀1 − m̀4)2 + m̀2m̀3 = δ2+ m̀2m̀3.
Since the Dirichlet and Neumann eigenvalues are roots of m̀2m̀3 the claimed
identity follows.
Next we describe the periodic spectrum specper(Q) of the operator Q =
Q1∂x +Q0 with domain given by the subspace of functions F in
Hper± :=
{
F ∈ H1
loc(R,C4) : ∀x ∈ R F (x+ 1) = ±F (x)
}
.
It coincides with the periodic spectrum of the spectral problem (2.2). Hence a
complex number λ ∈ C∗ is in specper(Q) iff M̀(λ, v) has an eigenvalue ±1. Since
det(M̀) = 1, the eigenvalues ξ± of M̀(λ) ≡ M̀(λ, v) satisfy
0 = det(ξ±I − M̀(λ)) = ξ2
± − 2∆(λ)ξ± + 1 (3.12)
and thus are given by
ξ± = ∆(λ)±
√
∆2(λ)− 1. (3.13)
Note that in (3.13), ξ+ and ξ− are determined up to the choice of a branch of√
∆2(λ)− 1.
On Spectral Properties of the L Operator. . . 479
Theorem 3.9. The periodic spectrum of Q(v) with v ∈ H1
c is discrete and
coincides with the zero set {λ ∈ C∗ : χp(λ, v) = 0} of the function χp(λ, v) :=
∆2(λ, v) − 1. Furthermore, the multiplicity of any root of χp coincides with its
algebraic multiplicity as a periodic eigenvalue. By Lemma 2.14 (i) and (ii) the
periodic spectrum is invariant under the involution λ→ −λ and for any periodic
eigenvalue of Q(q, p), 1
16λ is a periodic eigenvalue of Q(−q, p).
Proof. Let v ∈ H1
c be given. By (2.2), for any λ ∈ C∗ and F ∈
H1
loc(R,C4), the identity QF = λF is equivalent to F = (f, λ−1Bf) where f(x) =
M(x, λ, v)f(0). Hence the existence of a solution F of QF = λF with F (1) =
±F (0) is equivalent to ±1 being an eigenvalue of M(1, λ, v). By (3.13), ±1 is an
eigenvalue of M(1, λ, v) iff ∆(λ, v) = ±1. This proves the characterization. The
statement on the algebraic multiplicity of periodic eigenvalues is proved as the
corresponding one for the Dirichlet eigenvalues (cf., Lemma 3.5) and hence we
omit its proof.
For v = 0 the periodic spectrum of Q(v) can be computed explicitly. By
Lemma 2.16, χp(λ, 0) = cos2(ω(λ)) − 1 = − sin2(ω(λ)), where we recall that
ω(λ) = λ− 1
16λ .
Corollary 3.10. The periodic eigenvalues of Q(v) for v = 0 are{
nπ
2
(
+
√
1 +
1
4n2π2
+ 1
)
,
nπ
2
(
+
√
1 +
1
4n2π2
− 1
)
: n 6= 0
}⋃{
1
4
,−1
4
}
.
Each eigenvalue has algebraic multiplicity two.
It is convenient to list the two sequences of periodic eigenvalues of Q(0) with
their algebraic multiplicities as follows:
0 < · · · < λ−−1 = λ+
−1 < λ−0 = λ+
0 =
1
4
< λ−1 = λ+
1 < λ−2 = λ+
2 < · · ·
· · · < −λ−1 = −λ+
1 < −λ−0 = −λ+
0 = −1
4
< −λ−−1 = −λ+
−1 < · · · < 0.
We note that
∆(λ+
k , 0) = ∆(λ−k , 0) = (−1)k, k ∈ Z, and λ+
−k =
1
16λ+
k
, k ≥ 0. (3.14)
The periodic spectrum of Q(v) for arbitrary v ∈ H1
c is asymptotically close
to the one of Q(0).
Lemma 3.11 (Counting Lemma). For each potential in H1
c , there exist a
neighborhood U in H1
c and an integer N > 0 such that for every v ∈ U , the entire
function χp(λ, v) has exactly two roots in each of the domains Dn, −Dn, D−n,
and −D−n with n > N and exactly 4+8N roots in the annulus AN , counted with
their multiplicities. There are no further roots.
480 Thomas Kappeler and Yannick Widmer
Proof. By Lemma 2.17, χp(λ, v) = χp(λ, 0) (1 + o(1)) for |λ| → ∞ with λ 6∈⋃
n≥1Dn ∪ (−Dn), locally uniformly in v ∈ H1
c . Hence, for any potential in H1
c
there is a neighborhood U and an integer N ≥ 1 such that for any v ∈ U
|χp(λ, v)− χp(λ, 0)| < |χp(λ, 0)|, (3.15)
|χp(λ,−q, p)− χp(λ, 0, 0)| < |χp(λ, 0, 0)| (3.16)
on the boundaries of the discs Dn,−Dn, and Bn (defined in (3.2)) for any n ≥
N . The estimate (3.16) implies by Lemma 2.14 that∣∣∣∣χp( 1
16λ
, q, p
)
− χp
(
1
16λ
, 0, 0
)∣∣∣∣ = |χp(λ,−q, p)− χp(λ, 0, 0)|
< |χp(λ, 0, 0)| =
∣∣∣∣χp( 1
16λ
, 0, 0
)∣∣∣∣ .
It then follows that for any n ≥ N , (3.15) holds on the boundaries of ±Dn,±D−n
and Bn, B−n and hence on the boundary of An = Bn\B−n. By Rouché’s theorem,
χp(·, v) then has as many roots inside any of the discs ±Dn,±D−n and annuli An
as χp(λ, 0) for any n ≥ N . Since (An)n≥N covers C∗ the same argument shows
that χp(λ, v) has no roots in C∗\
(
AN ∪
⋃
n>N (Dn∪(−Dn)∪D−n∪(−D−n))
)
.
For v ∈ H2
c one can provide a bound N of Lemma 3.11 in terms of ‖v‖2. By
Lemma 2.14 (i) it is enough to consider the part of the periodic spectrum of Q(v)
in the half plane C+and by Lemma 3.11 the periodic eigenvalues in C+, counted
with their algebraic multiplicities, can be listed as a bi-infinite sequence
0 � · · · � λ−−1 � λ
+
−1 � λ
−
0 � λ
+
0 � λ
−
1 � λ
+
1 � · · · . (3.17)
Note that the segment {tv ∈ 1 : t ∈ [0, 1]} connecting v to 0 in H1
c is compact
and hence the integer N of Lemma 3.11 can be chosen uniformly in 0 ≤ t ≤
1. Furthermore, for any |k| ≥ N , ∆(λ+
k (tv), tv) = ∆(λ−k (tv), tv) and its sign is
constant in t. We conclude that for such k , ∆(λ±k , v) = (−1)k.
We finish this section by a discussion on the roots of ∆̇(λ, v) ≡ ∂λ∆(λ, v).
Since ∆ is even with respect to the variable λ, ∆̇ is odd and hence it is again
enough to look at the roots of ∆̇ in C+. For v = 0 one has ∆(λ) ≡ ∆(λ, 0) =
cos(ω(λ)), where ω(λ) = λ− 1
16λ , and hence
∆̇(λ) ≡ ∆̇(λ, 0) = −
(
1 +
1
16λ2
)
sin(ω(λ)). (3.18)
The roots of ∆̇(λ) in C+are given by the set of complex numbers consisting of
the bi-infinite sequence λ̇k ≡ λ̇k(0) = λ+
k (0), k ∈ Z, and the additional root
λ̇∗ = i
4 . Each of these roots has multiplicity one. By Lemma 2.14 (ii), one has
− 1
16λ2 ∆̇
(
1
16λ , q, p
)
= ∆̇(λ,−q, p). Since − 1
16λ2 ∆̇
(
1
16λ , q, p
)
and ∆̇
(
1
16λ , q, p
)
have
the same roots in C∗ (counted with their multiplicities), we can use the same
arguments as for the periodic eigenvalues of Q(v), to prove the following:
On Spectral Properties of the L Operator. . . 481
Lemma 3.12 (Counting Lemma). Given any potential in H1
c , there exists a
neighborhood U of it in H1
c and N > 0 (U and N can be chosen as in Lemma
3.11) so that for any v ∈ U , the function λ 7→ ∆̇(λ, v) has exactly one root in
each of the domains Dn, −Dn, D−n, and −D−n with n > N and 4 + 4N roots
in the annulus AN . There are no other roots.
By this lemma, the roots of ∆̇(·, v) in C+ \AN , counted with their algebraic
multiplicities, can be listed as a bi-infinite sequence
0 � · · · � λ̇−N−2 � λ̇−N−1 � λ̇N+1 � λ̇N+2 � · · · , λ̇k ∈ Dk, |k| > N (3.19)
such that any remaining root λ̇ in C+satisfies λ̇−N−1 � λ̇ � λ̇N+1.
We finish this section by establishing estimates for the periodic, Dirichlet,
and Neumann eigenvalues of Q(v). A first result concerns a priori bounds of the
imaginary part of any of these eigenvalues.
Lemma 3.13. For any v ∈ H2
c and any periodic, Dirichlet, or Neumann
eigenvalue λ ∈ C+,
| Imλ| ≤ ‖v‖2 + e‖q‖1 .
Proof. Let v ∈ H2
c and recall that Q = Q1∂x +Q0 with Q1, Q0 given by (1.4)
and for any F,G ∈ H1([0, 1],C4),
〈Q(v)F,G〉 = [Q1F ·G]10 + 〈F,Q(v)G〉,
where 〈·, ·〉 denotes the L2 inner product,
〈F,G〉 =
∫ 1
0
F (x) ·G(x)dx.
On the domains (contained in H1([0, 1],C4)) of Q, corresponding to periodic,
Dirichlet, or Neumann boundary conditions one has [Q1F · G]10 = 0. In partic-
ular if λ is a periodic, Dirichlet, or Neumann eigenvalue and F a corresponding
eigenfunction with 〈F, F 〉 = 1 one has
2i Imλ = λ− λ = 〈Q(v)F, F 〉 − 〈F,Q(v)F 〉 = 〈(Q(v)−Q(v))F, F 〉. (3.20)
Note that Q(v)−Q(v) = 2i ImQ0(v) and hence, by the Cauchy–Schwarz inequal-
ity and the normalization condition 〈F, F 〉 = 1,
|〈(Q(v)−Q(v))F, F 〉| ≤ ‖2(ImQ0(v))F‖L2 ,
where by (1.4) and (2.23)
‖2(ImQ0(v))F‖L2 ≤
1
2
(‖ImPp+ qx‖L∞+max
±
∥∥∥Im e±q/2
∥∥∥
L∞
) ≤ ‖v‖2+e‖q‖1 .
Note that µm, νm, λ̇m are close to mπ for m→∞. Our next aim is to obtain
more precise asymptotics for these quantities. First we need to establish the
following auxiliary result.
482 Thomas Kappeler and Yannick Widmer
Lemma 3.14. For any bi-infinite sequence of complex numbers (ζn)n ⊂ C∗
satisfying ζn = nπ +O(1) as n→ ±∞ one has
∆|λ=ζn
= cos(ζn) + `2n, ∆̇
∣∣∣
λ=ζn
= − sin(ζn) + `2n,
δ|λ=ζn
= `2n, δ̇
∣∣∣
λ=ζn
= `2n,
χD|λ=ζn
= − sin(ζn) + `2n, χ̇D|λ=ζn
= − cos(ζn) + `2n.
These estimates hold uniformly on subsets of sequences (ζn)n where (ω(ζn)−nπ)n
is bounded. If in fact ζn = nπ + `2n, then
sin(ζn) = `2n and cos(ζn) = (−1)n + `2n,
yielding in particular the sharper asymptotics
∆|λ=ζn
= (−1)n + `2n and ∆̇
∣∣∣
λ=ζn
= `2n for n→ ±∞.
Proof. The stated asymptotics follow from Theorem 2.12 (iii) and (iv).
Lemma 3.15. For any v ∈ H1
c , the roots of ∆̇ in C+have the following
asymptotics as n→∞
λ̇n = nπ + `2n,
1
16λ̇−n
= nπ + `2n.
These estimates hold locally uniformly on H1
c .
Proof. Since by Lemma 3.12, λ̇n = nπ + O(1), it follows from Lemma 3.14
that
0 = ∆̇(λ̇n) = − sin(λ̇n) + `2n (3.21)
or sin(λ̇n) = `2n. Since by Lemma 3.12, |λ̇n− nπ| < π/3 for |n| large enough, one
has ∣∣∣λ̇n − nπ∣∣∣ cos(π/3) ≤
∣∣∣λ̇n − nπ∣∣∣ ∣∣∣∣∫ 1
0
cos
((
λ̇n − nπ
)
s+ nπ
)
ds
∣∣∣∣
=
∣∣∣sin (λ̇n)− sin(nπ)
∣∣∣ = `2n,
proving the first claimed asymptotics. They in turn yield the second ones by
Lemma 2.14 (ii).
By the same arguments one can prove that similar results hold for the Dirichlet
eigenvalues.
Lemma 3.16. For any v ∈ H1
c , the Dirichlet eigenvalues of Q(v) in C+have
the following asymptotics as n→∞
µn = nπ + `2n and
1
16µ−n
= nπ + `2n.
These estimates hold locally uniformly in H1
c .
On Spectral Properties of the L Operator. . . 483
We will now use Lemma 3.15 and Lemma 3.16 to prove the following result
for the periodic eigenvalues of Q(v).
Lemma 3.17. For any v ∈ H1
c , the periodic eigenvalues of Q(v) in C+have
the following asymptotics as n→∞
λ±n = nπ + `2n and
1
16λ±−n
= nπ + `2n.
These estimates hold locally uniformly on H1
c .
Proof. Let v ∈ H1
c be given. Since by Lemma 3.16, µn = nπ + `2n, Lemma
3.14 yields δ(µn) = `2n. Hence by Lemma 3.8 one has ∆2(µn)− 1 = δ2(µn) = `1n.
On the other hand, Lemma 3.14 also yields that ∆(µn) = (−1)n + `2n. Writing
∆2(µn) − 1 = (∆(µn) − 1)(∆(µn) + 1) the two latter estimates together imply
that
∆(µn) = (−1)n + `1n. (3.22)
A similar estimate holds for ∆
(
λ̇n
)
. Indeed, since λ̇n − µn = `2n by Lemma 3.15
and Lemma 3.16,
∆
(
λ̇n
)
−∆(µn) =
(
λ̇n − µn
) ∫ 1
0
∆̇
(
tλ̇n + (1− t)µn
)
dt,
and ∆̇
(
tλ̇n + (1 − t)µn
)
= `2n uniformly in 0 ≤ t ≤ 1 by Lemma 3.14, it follows
that ∆
(
λ̇n
)
−∆(µn) = `1n. Together with (3.22) this yields ∆
(
λ̇n
)
= (−1)n + `1n.
The latter estimates can be applied as follows. Since ∆̇
(
λ̇n
)
= 0 one has
[
(1 −
t)∆̇
(
tλ±n + (1− t)λ̇n
)]1
0
= 0 and therefore integrating by parts,
∆
(
λ±n
)
−∆
(
λ̇n
)
=
(
λ±n − λ̇n
) ∫ 1
0
∆̇
(
tλ±n + (1− t)λ̇n
)
dt
=
(
λ±n − λ̇n
)2 ∫ 1
0
(1− t)∆̈
(
tλ±n + (1− t)λ̇n
)
dt.
Hence (
λ±n − λ̇n
)2 ∫ 1
0
(1− t)∆̈
(
tλ±n + (1− t)λ̇n
)
dt = `1n. (3.23)
Since ∆ is analytic in λ and ∆̇(ζn) = − sin(ζn) + `2n by Lemma 3.14, Cauchy’s
estimate yields
∆̈
(
tλ±n + (1− t)λ̇n
)
= − cos
(
tλ±n + (1− t)λ̇n
)
+ `2n
uniformly in 0 ≤ t ≤ 1. For n sufficiently large, tλ±n + (1 − t)λ̇n is in Dn and
hence ∫ 1
0
(1− t)∆̈
(
tλ±n + (1− t)λ̇n
)
dt
is uniformly bounded away from zero for such n. So (3.23) yields λ±n − λ̇n = `2n.
Since by Lemma 3.15, λ̇n = nπ+ `2n, the first claimed asymptotics follow. Those
then yield the second ones by Lemma 2.14 (ii).
484 Thomas Kappeler and Yannick Widmer
4. Proofs of Theorem 1.1 and Theorem 1.2
In this section we prove Theorem 1.1 and Theorem 1.2. Actually, we prove
stronger versions of them as stated in Theorem 4.10 and Theorem 4.15 below.
For any potential v in H1
c let
Gn ≡ Gn(v) := [λ−n , λ
+
n ] :=
{
(1− t)λ−n + tλ+
n : 0 ≤ t ≤ 1
}
, n ∈ Z . (4.1)
By a slight abuse of terminology we refer to Gn as the nth closed spectral gap,
although if v is not real valued it lacks a spectral interpretation. Furthermore for
any v ∈ H1
c and n ∈ Z we introduce the gap length
γn(v) := λ+
n (v)− λ−n (v), n ∈ Z. (4.2)
Note that in general, γn(v) is a complex number, but in case v is real valued, it
is real and equals the length of the gap Gn(v). For d ≥ 1 and s ∈ R≥0, denote
by Hs(T2,Cd) the Sobolev space of order s of two periodic functions with values
in Cd,
Hs(T2,Cd) :=
{
u =
∑
n∈Z
unen : un ∈ Cd and ‖u‖s <∞
}
,
‖u‖s :=
(∑
n∈Z
〈n〉2s|un|2
)1/2
,
where T2 = R/2Z, en(x) = einπx, and |a| =
(∑d
j=1 |aj |2
)1/2
for any a =
(a1, . . . , ad) ∈ Cd. We recall that the weights 〈n〉s :=
(
1 + π2n2
)s/2
are sub-
multiplicative for any s ≥ 0, i.e., 〈n+m〉s ≤ 〈n〉s〈m〉s. The L2-inner product is
defined for f, g ∈ H0(T2,Cd) ≡ L2(T2,Cd) by
〈f, g〉c =
1
2
∫ 2
0
fg dx =
1
2
∫ 2
0
d∑
j=1
f (j)(x)g(j)(x) dx, (4.3)
where f (j), 1 ≤ j ≤ d, denote the components of f . For a scalar valued function
u ∈ Hs(T2,C) and a vector valued function v ∈ Hs(T2,Cd) with v =
∑
n∈Z vnen
and vn ∈ Cd one has
‖uv‖s ≤ ‖u‖s
∑
n∈Z
〈n〉s|vn|. (4.4)
Hence by the Cauchy–Schwarz inequality,
∑
n∈Z
1
〈n〉2 ≤ 1+ 2
π2
∑∞
n=1
1
n2 , and since
∞∑
n=1
1
n2
=
π2
6
, (4.5)
one has
‖uv‖s ≤ 2 ‖u‖s ‖v‖s+1 . (4.6)
On Spectral Properties of the L Operator. . . 485
Recall that by (2.5), (2.6), Q(q, p) = Q1∂x +Q0(q, p) with
Q1 =
(
R
)
, Q0(q, p) =
(
A(q, p) B(q, p)
B(q, p)
)
and
A(q, p) =
1
4
ϕJ, ϕ := −i(Pp+ qx), B(q, p) =
1
4
(
cosh(q/2) − sinh(q/2)
− sinh(q/2) cosh(q/2)
)
.
It turns out to be useful to introduce the linear isomorphism Hs+1
C × Hs+1
C →
Hs+1
C ×Hs
C, (q, p) 7→ (q, ϕ) and use (q, ϕ) instead of (q, p) as phase space variables.
Furthermore introduce
H̃s+1
c := Hs+1
C ×Hs
C (4.7)
and by a slight abuse of notation we write Q ≡ Q(q, ϕ) and Q0 ≡ Q0(q, ϕ) for
the operators Q(q, p) and Q0(q, p) respectively. We rewrite equation (2.11) in the
form
Q(λ)F = Q0F, Q(λ) := −Q1∂x + λI (4.8)
and introduce the L2-orthogonal basis of L2(T2,C4),
e(j)
n (x) = en(x)a(j), en(x) = einπx, ∀ 1 ≤ j ≤ 4, n ∈ Z,
where a(1), a(2), a(3), a(4) denote the standard basis in C4, a(1) = (1, 0, 0, 0), a(2) =
(0, 1, 0, 0), a(3) = (0, 0, 1, 0), a(4) = (0, 0, 0, 1). Note that
Q(λ)e(1)
n = (λ+ nπ)e(1)
n , Q(λ)e(2)
n = (λ− nπ)e(2)
n , (4.9)
Q(λ)e(3)
n = λe(3)
n , Q(λ)e(4)
n = λe(4)
n , (4.10)
suggesting to decompose Hs(T2,C4) with s ≥ 0 for any given n ∈ Z as
Hs(T2,C4) = Pn ⊕Kn, where
Pn :=
{
f
(1)
−ne
(1)
−n + f (2)
n e(2)
n : f
(1)
−n, f
(2)
n ∈ C
}
,
Kn :=
∑
k∈Z,1≤j≤4
f
(j)
k e
(j)
k ∈ H
s(T2,C4) : f
(j)
k ∈ C, f (1)
−n = 0, f (2)
n = 0
.
Denote the L2-orthogonal projections onto Pn and Kn by Pn and Kn, respectively.
The subspaces Pn and Kn are invariant under Q(λ). Furthermore, introduce for
any n ∈ Z the complex strip
Πn = {λ ∈ C : |Rλ− nπ| ≤ π/2} . (4.11)
Note that these strips cover C and that for any n 6= 0, the restriction of Q(λ) to
Kn, again denoted by Q(λ), is invertible for any λ ∈ Πn. Writing F = u+ v with
u := PnF and v := KnF , equation (4.8) decomposes into the following system of
equations
Q(λ)u = PnQ0(u+ v) , (4.12)
486 Thomas Kappeler and Yannick Widmer
Q(λ)v = KnQ0(u+ v) , (4.13)
called P - and, respectively, K-equation. (Note that in this section, u and v have
a different meaning than elsewhere in the paper.) Given any n 6= 0 we first solve
the K-equation for any given u ∈ Pn and then substitute the solution into the
P -equation, leading to a 2× 2 system of linear equations with a 2× 2 coefficient
matrix Sn, which is singular precisely when λ is a periodic eigenvalue of Q.
Actually, to solve (4.12), it suffices to determine Q0v. Hence in a first step,
we derive from (4.13) an equation for Q0v instead of v. Once u and Q0v are
found, v can be easily determined from (4.13), v = Q(λ)−1Kn(Q0u +Q0v). We
begin by deriving from (4.13) an equation for Q0v. Given any n 6= 0 and λ ∈ Πn,
apply the operator Q0Q(λ)−1 to (4.13) to obtain
Q0v = Q0Q(λ)−1KnQ0(u+ v), (4.14)
which leads to the following equation for ṽ = Q0v ∈ L2(T2,C4),
(Id− Tn)ṽ = TnQ0u, Tn := Q0Q(λ)−1Kn : L2(T2,C4)→ L2(T2,C4).
We then prove that for |n| sufficiently large, T 4
n is a contraction implying that
for such n, Id − T 4
n is invertible. The invertibility of the operator Id − Tn then
follows from the identity
(Id− Tn)−1 = (Id+ Tn)(Id+ T 2
n)(Id− T 4
n)−1.
First we need to introduce some more notation. For d = 1, 2, 4 and s ≥ 0 we
consider on Hs(T2,Cd) the shifted norms
‖u‖s;n := ‖uen‖s =
(∑
k∈Z
〈k + n〉2s|un|2
)1/2
.
Note that the estimate (4.6) continues to be valid for these norms. More precisely,
the following holds:
Lemma 4.1. Let s ∈ R≥0 and n ∈ Z. Then the following holds:
(i) For any u ∈ Hs(T2,C), v ∈ Hs+1(T2,Cd):
‖uv‖s;n ≤ 2 ‖u‖s;n ‖v‖s+1 , ‖uv‖s;n ≤ 2 ‖u‖s ‖v‖s+1;n .
(ii) For any u ∈ Hs+1(T2,C), v ∈ Hs(T2,Cd):
‖uv‖s;n ≤ 2 ‖u‖s+1;n ‖v‖s , ‖uv‖s;n ≤ 2 ‖u‖s+1 ‖v‖s;n .
Proof. One computes
‖uv‖s;n = ‖uven‖s ≤ 2 ‖uen‖s ‖v‖s+1 = 2 ‖u‖s;n ‖v‖s+1 .
The other inequalities are obtained in a similar fashion.
On Spectral Properties of the L Operator. . . 487
Lemma 4.2. For any (q, ϕ) ∈ H̃s+1
c with s ≥ 0, l ∈ Z, and λ ∈ Πn with n 6=
0, the following holds:
(i) Decomposing
Tn = Q0Q(λ)−1Kn : (Hs(T2,C4), ‖·‖s;l)→ (Hs(T2,C4), ‖·‖s;l)
according to
Q0 =
(
A
)
+
(
B
)
+
(
B
)
,
the resulting operators satisfy∥∥∥∥(A )
Q(λ)−1Kn
∥∥∥∥
s;l
≤ ‖ϕ‖s , (4.15)∥∥∥∥(B
)
Q(λ)−1Kn
∥∥∥∥
s;l
≤ ‖sinh(q/2)‖s + ‖cosh(q/2)‖s , (4.16)∥∥∥∥( B
)
Q(λ)−1Kn
∥∥∥∥
s;l
≤
‖sinh(q/2)‖s+1 + ‖cosh(q/2)‖s+1
|n|
. (4.17)
(ii) Tn is bounded. More precisely, ‖Tn‖s;l ≤ Rs, where
Rs ≡ Rs(q, ϕ) := ‖ϕ‖s + ‖sinh(q/2)‖s+1 + ‖cosh(q/2)‖s+1 . (4.18)
Proof. Let n 6= 0. Clearly (4.18) follows from (4.15)–(4.17). The latter
estimates are proved separately. To prove (4.15) note that for m 6= n
min
λ∈Πn
|λ−mπ| ≥ |n−m| ≥ 1, (4.19)
implying that for any λ ∈ Πn, the restriction of Q(λ) to the invariant subspace
Kn is invertible and that its inverse is uniformly bounded for λ ∈ Πn. For any
F =
∑
m∈Z,j=1,2,3,4 f
(j)
m e
(j)
m with f
(j)
m ∈ C, and λ ∈ Πn
Q(λ)−1KnF =
∑
m 6=n
1
λ−mπ
(
f
(1)
−me
(1)
−m + f (2)
m e(2)
m
)
+
1
λ
∑
m∈Z
(
f (3)
m e(3)
m + f (4)
m e(4)
m
)
is well defined. Taking into account Lemma 4.1 and definition (2.7) of A one has∥∥∥∥(A )
Q(λ)−1KnF
∥∥∥∥
s;l
=
∥∥∥∥∥∥
(
A
)∑
m6=n
1
λ−mπ
(
f
(1)
−me
(1)
−m + f (2)
m e(2)
m
)∥∥∥∥∥∥
s;l
≤1
4
‖ϕ‖s
∑
m 6=n
∣∣∣f (1)
−m+l
∣∣∣ 〈−m+ l〉s +
∣∣∣f (2)
m+l
∣∣∣ 〈m+ l〉s
|n−m|
.
For 1 ≤ j ≤ 4, let f (j) :=
∑
m∈Z f
(j)
m e
(j)
m . By the Cauchy–Schwarz inequality and
(4.5) one then concludes
488 Thomas Kappeler and Yannick Widmer
∑
m6=n
∣∣∣f (1)
−m+l
∣∣∣ 〈−m+ l〉s +
∣∣∣f (2)
m+l
∣∣∣ 〈m+ l〉s
|n−m|
≤
∥∥∥f (1) + f (2)
∥∥∥
s;l
∑
m 6=n
1
|m− n|2
1/2
≤ π√
3
‖F‖s;l ,
where we used that∥∥∥f (1) + f (2)
∥∥∥2
s;l
=
∥∥∥f (1)
∥∥∥2
s;l
+
∥∥∥f (2)
∥∥∥2
s;l
≤ ‖F‖2s;l .
This shows (4.15). A similar estimate holds for∥∥∥∥(B
)
Q(λ)−1KnF
∥∥∥∥
s;l
,
where 1
4 ‖ϕ‖s in the estimate above is replaced by 1
4 ‖sinh(q/2)‖s+
1
4 ‖cosh(q/2)‖s.
This yields estimate (4.16). On the other hand,(
B
)
Q(λ)−1KnF =
(
B
)
1
λ
∑
m∈Z
(
f (3)
m e(3)
m + f (4)
m e(4)
m
)
.
Since by (4.19), 1
|λ| ≤
1
|n| for any λ ∈ Πn, estimate (4.17) then follows from
Lemma 4.1.
Lemma 4.3. Let (q, ϕ) ∈ H̃s+1
c with s ≥ 0 and λ ∈ Πn with n ∈ Z \ {0}.
Then the following holds:
(i) For any F =
∑4
j=1 f
(j) ∈ Hs(T2,C4),∥∥∥∥Tn(A )
Q(λ)−1Knf
(1)
∥∥∥∥
s;−n
≤ 1
2|n|
Rs(q, ϕ) ‖ϕ‖s
∥∥∥f (1)
∥∥∥
s;−n
(4.20)
and ∥∥∥∥Tn(A )
Q(λ)−1Knf
(2)
∥∥∥∥
s;n
≤ 1
2|n|
Rs(q, ϕ) ‖ϕ‖s
∥∥∥f (2)
∥∥∥
s;n
, (4.21)
while for j = 3, 4
Tn
(
A
)
Q(λ)−1Knf
(j) = 0.
(ii) Furthermore ∥∥∥∥∥
[(
A
)
Q(λ)−1Kn
]3
∥∥∥∥∥
s;±n
≤ 1
|n|
‖ϕ‖3s . (4.22)
On Spectral Properties of the L Operator. . . 489
(iii)For any F =
∑4
j=1 f
(j) ∈ Hs(T2,C4),∥∥∥∥Tn(A )
Q(λ)−1Kn
∥∥∥∥
s;±n
≤ Rs(q, ϕ)
(
2√
|n|
‖ϕ‖s +Rs;|n|(ϕ)
)
, (4.23)
where for any g =
∑
k∈Z gkek ∈ Hs(T2,C),
Rs;|n|(g) :=
∑
|k|≥|n|
〈k〉2s|gk|2
1/2
. (4.24)
Proof. (i) Writing ϕ =
∑
m∈Z ϕmem, it is easy to see that for F =
∑4
j=1 f
(j) ∈
Hs(T,C4)
g(1) :=Q(λ)−1Kn
(
A
)
Q(λ)−1Knf
(2)
=
1
4
∑
k 6=−n
∑
m6=n
f
(2)
m ϕk−m
(λ−mπ)(λ+ kπ)
e
(1)
k (4.25)
and similarly
g(2) :=Q(λ)−1Kn
(
A
)
Q(λ)−1Knf
(1)
=
1
4
∑
k 6=n
∑
m 6=−n
−f (1)
m ϕk−m
(λ+mπ)(λ− kπ)
e
(2)
k , (4.26)
while for j = 3, 4 (
A
)
Q(λ)−1Knf
(j) = 0. (4.27)
Note that the coefficients of g(1) =
∑
k 6=−n g
(1)
k e
(1)
k and g(2) =
∑
k 6=n g
(2)
k e
(2)
k are
given by
g
(1)
k =
1
4
∑
m 6=n
f
(2)
m ϕk−m
(λ−mπ)(λ+ kπ)
, g
(2)
k = −1
4
∑
m6=−n
f
(1)
m ϕk−m
(λ+mπ)(λ− kπ)
.
By Lemma 4.1, one has for j = 1, 2 and i = 1, 2 such that {1, 2} = {i, j}∥∥∥∥Tn(A )
Q(λ)−1Knf
(j)
∥∥∥∥
s;l
=
∥∥Q0g
(i)
∥∥
s;l
≤ 1
4
(‖ϕ‖s + ‖sinh(q/2)‖s + ‖cosh(q/2)‖s)
∥∥∥g(i)el
∥∥∥
W s,1
,
where ∥∥∥g(i)
∥∥∥
W s,1
:=
∑
m∈Z
〈m〉s
∣∣∣g(i)
m
∣∣∣ . (4.28)
490 Thomas Kappeler and Yannick Widmer
The bounds (4.20) and (4.21) then follow from corresponding bounds of∥∥g(1)en
∥∥
W s,1 and
∥∥g(2)e−n
∥∥
W s,1 . Indeed,
∥∥∥g(1)en
∥∥∥
W s,1
=
∥∥∥∥∥∥1
4
∑
k 6=−n
∑
m6=n
f
(2)
m ϕk−m
(λ−mπ)(λ+ kπ)
e
(1)
k en
∥∥∥∥∥∥
W s,1
≤ 1
4
∑
k 6=−n
∑
m6=n
〈k + n〉s
|n−m||n+ k|
∣∣∣f (2)
m
∣∣∣ |ϕk−m| . (4.29)
Since for k 6= −n, 〈k+n〉s
|k+n| ≤ 4〈k + n〉s−1 and 〈k + n〉s−1 ≤ 〈m+ n〉s−1〈k −m〉s−1
one obtains by Young’s inequality
1
4
∑
k 6=−n
∑
m6=n
〈k + n〉s
|n−m||n+ k|
∣∣∣f (2)
m
∣∣∣ |ϕk−m| ≤ ∑
m 6=n
∣∣∣f (2)
m
∣∣∣ 〈m+ n〉s−1
|n−m|
∑
k∈Z
〈k〉s−1|ϕk|
and by the Cauchy–Schwarz inequality,
∑
m6=n
|f (2)
m |〈m+ n〉s−1
|n−m|
=
∑
m 6=n
1
〈m+ n〉|n−m|
|f (2)
m |〈m+ n〉s
≤
∥∥∥f (2)
∥∥∥
s;n
∑
m6=n
1
〈m+ n〉2|n−m|2
1/2
. (4.30)
For n > 0,
∑
m6=n
1
|n−m|2〈m+n〉2 is bounded by
1
π2n2
+
1
π2n2
∑
m<0
1
m2
+
∑
m>0,m 6=n
1
|n−m|2π2n2
≤ 1
n2
(
1
π2
+
3
π2
∑
m>0
1
m2
)
.
(4.31)
Since a similar statement holds for n < 0 and
∑
k∈Z
〈k〉s−1|ϕk| ≤
(∑
k∈Z
1
〈k〉2
)1/2(∑
k∈Z
〈k〉2s|ϕk|2
)1/2
≤ ‖ϕ‖s
(
1 +
1
3
)1/2
as well as
∑∞
n=1
1
n2 = π2
6 one has∥∥∥g(1)en
∥∥∥
W s,1
≤ ‖ϕ‖s
2
|n|
∥∥∥f (2)
∥∥∥
s;n
. (4.32)
A similar computation yields∥∥∥g(2)e−n
∥∥∥
W s,1
≤ ‖ϕ‖s
2
|n|
∥∥∥f (1)
∥∥∥
s;−n
. (4.33)
This proves (4.21) and (4.20).
On Spectral Properties of the L Operator. . . 491
(ii) To prove (4.22), note that by (4.30) the definition (4.25) of g(1) and the
estimate (4.4)∥∥∥∥∥
[(
A
)
Q(λ)−1Kn
]2
f (2)
∥∥∥∥∥
s;±n
=
∥∥∥∥(A )
g(1)
∥∥∥∥
s;±n
≤ 1
4
‖ϕ‖s
∥∥∥g(1)e±n
∥∥∥
W s,1
and hence by (4.32)∥∥∥∥∥
[(
A
)
Q(λ)−1Kn
]2
f (2)
∥∥∥∥∥
s;±n
≤ 1
2|n|
‖ϕ‖2s
∥∥∥f (2)
∥∥∥
s;±n
. (4.34)
It then follows by (4.15) that∥∥∥∥∥
[(
A
)
Q(λ)−1Kn
]3
f (2)
∥∥∥∥∥
s;±n
≤ ‖ϕ‖s
∥∥∥∥∥
[(
A
)
Q(λ)−1Kn
]2
f (2)
∥∥∥∥∥
s;±n
≤ 1
2|n|
‖ϕ‖3s
∥∥∥f (2)
∥∥∥
s;±n
. (4.35)
By the definition of A = 1
4ϕJ , all components of
(
A
)
Q(λ)−1Knf
(1) vanish
except the second one. Therefore (4.34) implies∥∥∥∥∥
[(
A
)
Q(λ)−1Kn
]3
f (1)
∥∥∥∥∥
s;±n
≤ 1
2|n|
‖ϕ‖2s
∥∥∥∥(A )
Q(λ)−1Knf
(1)
∥∥∥∥
s;±n
.
By (4.15) it then follows that∥∥∥∥∥
[(
A
)
Q(λ)−1Kn
]3
f (1)
∥∥∥∥∥
s;±n
≤ 1
2|n|
‖ϕ‖3s
∥∥∥f (1)
∥∥∥
s;±n
.
In view of (4.27) the claimed estimate (4.22) then follows.
(iii) In view of item (i) and the definitions (4.25) and (4.26) of g(1) and g(2),
it remains to bound
∥∥g(2)en
∥∥
W s,1 and
∥∥g(1)e−n
∥∥
W s,1 . One computes
∥∥∥g(2)en
∥∥∥
W s,1
=
∥∥∥∥∥∥1
4
∑
k 6=n
∑
m6=−n
−f (1)
m ϕk−m
(λ+mπ)(λ− kπ)
e
(2)
k en
∥∥∥∥∥∥
W s,1
≤ 1
4
∑
k 6=n
∑
m 6=−n
∣∣∣f (1)
m
∣∣∣ |ϕk−m|〈k + n〉s
|n+m||n− k|
.
Since 〈k + n〉s ≤ 〈k −m〉s〈m+ n〉s, one has
∑
k 6=n
∑
m6=−n
∣∣∣f (1)
m
∣∣∣ |ϕk−m|〈k + n〉s
|n+m||n− k|
≤
∑
k 6=n
∑
m 6=−n
∣∣∣f (1)
m
∣∣∣ 〈m+ n〉s|ϕk−m|〈k −m〉s
|n+m||n− k|
.
492 Thomas Kappeler and Yannick Widmer
We split the latter sum into three parts defined by the three sets of summation
indices
{(k,m) : |n− k| > |n|/2} , {(k,m) : |n− k| ≤ |n|/2, |n+m| > |n|/2} ,
and
{(k,m) : |n− k| ≤ |n|/2, |n+m| ≤ |n|/2} .
Let
I :=
∑
|n−k|> |n|
2
∑
m6=−n
|f (1)
m |〈m+ n〉s|ϕk−m|〈k −m〉s
|n+m||n− k|
.
Then by the Cauchy–Schwarz inequality I is bounded by
C
∑
|n−k|> |n|
2
∑
m 6=−n
|f (1)
m |2〈m+ n〉2s|ϕk−m|2〈k −m〉2s
1/2
,
where
C :=
∑
|n−k|> |n|
2
∑
m6=−n
1
|n− k|2|n+m|2
1/2
and hence
I ≤
∑
|n−k|> |n|
2
1
|n− k|2
1/2 ∑
m6=−n
1
|n+m|2
1/2
‖ϕ‖s
∥∥∥f (1)
∥∥∥
s;n
.
By a similar computation
II :=
∑
1≤|n−k|≤ |n|
2
∑
|n+m|> |n|
2
∣∣∣f (1)
m
∣∣∣ 〈m+ n〉s|ϕk−m|〈k −m〉s
|n+m||n− k|
≤
∑
1≤|n−k|≤ |n|
2
1
|n− k|2
1/2 ∑
|n+m|> |n|
2
1
|n+m|2
1/2
‖ϕ‖s
∥∥∥f (1)
∥∥∥
s;n
.
Since ∑
|l|> |n|
2
1
l2
≤ 2
∑
l>
|n|
2
1
l2
≤ 2
∫ ∞
|n|
2
1
x2
dx =
4
|n|
,
it then follows that
1
4
I +
1
4
II ≤ 1
2
(
2
π2
6
)1/2( 4
|n|
)1/2 ‖ϕ‖s ∥∥∥f (1)
∥∥∥
s;n
≤ 2√
|n|
‖ϕ‖s
∥∥∥f (1)
∥∥∥
s;n
.
On Spectral Properties of the L Operator. . . 493
Now let us turn to the sum
III :=
∑
1≤|n−k|≤ |n|
2
∑
1≤|n+m|≤ |n|
2
∣∣∣f (1)
m
∣∣∣ 〈m+ n〉s|ϕk−m|〈k −m〉s
|n+m||n− k|
.
Since |k−m| ≥ 2|n| − |k−n| − |m+n| ≥ |n| for k,m with |n− k| ≤ |n|2 and |m+
n| ≤ |n|2 , the sum III is bounded by ∑
1≤|n−k|≤ |n|
2
1
|n− k|2
1/2 ∑
1≤|n+m|≤ |n|
2
1
|n+m|2
1/2
×
∑
|k|≥|n|
|ϕk|2〈k〉2s
1/2 ∥∥∥f (1)
∥∥∥
s;n
≤ 2
π2
6
Rs;n(ϕ)
∥∥∥f (1)
∥∥∥
s;n
and hence
1
4
III ≤ Rs;n(ϕ)
∥∥∥f (1)
∥∥∥
s;n
.
Altogether, we thus have proved that∥∥∥g(2)en
∥∥∥
W s,1
≤ 2√
|n|
‖ϕ‖s
∥∥∥f (1)
∥∥∥
s;n
+Rs;|n|(ϕ)
∥∥∥f (1)
∥∥∥
s;n
. (4.36)
Combining (4.32) and (4.36) one obtains the estimate∥∥∥∥Tn(A )
Q(λ)−1Kn
∥∥∥∥
s;n
≤ Rs(q, ϕ)
(
2√
|n|
‖ϕ‖s +Rs;|n|(ϕ)
)
. (4.37)
Similarly, one shows that∥∥∥g(1)e−n
∥∥∥
W s,1
≤ 2√
|n|
‖ϕ‖s
∥∥∥f (2)
∥∥∥
s;−n
+Rs;|n|(ϕ)
∥∥∥f (2)
∥∥∥
s;−n
and deduces∥∥∥∥Tn(A )
Q(λ)−1Kn
∥∥∥∥
s;−n
≤ Rs(q, ϕ)
(
2√
|n|
‖ϕ‖s +Rs;|n|(ϕ)
)
. (4.38)
This proves (4.23).
Decomposing Tn as in Lemma 4.2, the following identities can be verified in
a straightforward way.
Lemma 4.4. Let (q, ϕ) ∈ H̃s+1
c with s ≥ 0 and λ ∈ Πn with n ∈ Z \ {0}.
Then (
A
)
Q(λ)−1Kn
(
B
)
Q(λ)−1Kn = 0,
494 Thomas Kappeler and Yannick Widmer(
B
)
Q(λ)−1Kn
(
A
)
Q(λ)−1Kn = 0,(
B
)
Q(λ)−1Kn
(
B
)
Q(λ)−1Kn = 0,(
B
)
Q(λ)−1Kn
(
B
)
Q(λ)−1Kn = 0.
Lemma 4.5. Let (q, ϕ) ∈ H̃s+1
c with s ≥ 0 and λ ∈ Πn with n ∈ Z \ {0}.
(i) There exists an absolute constant C0 ≥ 1 so that∥∥T 4
n
∥∥
s;±n ≤
C0
|n|
R4
s, Rs ≡ Rs(q, ϕ) := ‖ϕ‖s+‖cosh(q/2)‖s+1+‖sinh(q/2)‖s+1 .
(ii) With Rs,|n|(ϕ) given as in (4.24) one has
∥∥T 2
n
∥∥
s;±n ≤ Rs(q, ϕ)
(
‖sinh(q/2)‖s+1 + ‖cosh(q/2)‖s+1
|n|
+
2√
|n|
‖ϕ‖s +Rs;|n|(ϕ)
)
.
(iii)For any F =
∑4
j=1 f
(j) ∈ Hs(T2,C4) with f (1) = 0, the following sharper
estimate holds ∥∥T 2
nF
∥∥
s;n
≤ 1
|n|
R2
s(q, ϕ) ‖F‖s;n .
(iv) For any F =
∑4
j=1 f
(j) ∈ Hs(T2,C4) with f (2) = 0, one has
∥∥T 2
nF
∥∥
s;−n ≤
1
|n|
R2
s(q, ϕ) ‖F‖s;−n .
Remark 4.6. It follows from Lemma 4.5 (i) that T 4
n is a 1
2 -contraction for
|n| ≥ 2C0R
4
s. In contrast, the estimate of Lemma 4.5 (ii) implies that, T 2
n is
a 1
2 -contraction for |n| ≥ N where N can be chosen locally uniformly on H̃s+1
C .
Proof. (i) Decompose Tn into three terms,
Tn =
(
A
)
Q(λ)−1Kn +
(
B
)
Q(λ)−1Kn +
(
B
)
Q(λ)−1Kn.
By Lemma 4.4, T 4
n consists of a sum of terms, each containing(
B
)
Q(λ)−1Kn or
[(
A
)
Q(λ)−1Kn
]3
as a factor. Using (4.15), (4.17), and (4.22) one obtains the claimed estimate∥∥T 4
n
∥∥
s;n
≤ C0R
4
s/|n|.
On Spectral Properties of the L Operator. . . 495
(ii) Note that by Lemma 4.3
∥∥T 2
n
∥∥
s;±n ≤
∥∥∥∥Tn( B
B
)
Q(λ)−1Kn
∥∥∥∥
s;±n
+Rs(q, ϕ)
(
2√
|n|
‖ϕ‖s +Rs;n(ϕ)
)
and by Lemma 4.2 and Lemma 4.4,∥∥∥∥Tn( B
B
)
Q(λ)−1Kn
∥∥∥∥
s;±n
≤ Rs(q, ϕ)
‖sinh(q/2)‖s+1 + ‖cosh(q/2)‖s+1
|n|
. (4.39)
(iii) For any F =
∑4
j=1 f
(j) ∈ Hs(T2,C4)
∥∥T 2
nF
∥∥
s;n
≤
∥∥∥∥Tn( B
B
)
Q(λ)−1KnF
∥∥∥∥
s;n
+
∥∥∥∥Tn(A )
Q(λ)−1KnF
∥∥∥∥
s;n
.
If f (1) = 0 then by Lemma 4.3 (i)∥∥∥∥Tn(A )
Q(λ)−1KnF
∥∥∥∥
s;n
≤ 1
2|n|
Rs(q, ϕ) ‖ϕ‖s
∥∥∥f (2)
∥∥∥
s;n
.
Hence (4.39) yields (iii).
(iv) Arguing as in the proof of item (iii) one obtains (iv).
We now go back to the K- and P -equation. Let (q, ϕ) ∈ H̃s+1
c be given.
Instead of the K-equation (4.13) we consider (4.14) which by the definition of Tn
takes the form Q0v = TnQ0(u+ v). Solving for Q0v yields
(Id− Tn)Q0v = TnQ0u. (4.40)
By Lemma 4.5 (iv), T 4
n is a 1/2-contraction for any n with |n| ≥ 2C0R
4
s. It follows
that for such n, (Id − T 4
n) and hence (Id − Tn) are invertible where the inverse
of (Id− Tn) is given by
T̂n := (Id− Tn)−1 = (Id− T 4
n)−1(Id+ Tn + T 2
n + T 3
n).
By Lemma 4.2 for any s ≥ 0 and |n| ≥ 2C0R
4
s∥∥∥T̂n∥∥∥
s;±n
≤ 2(1 +Rs +R2
s +R3
s) ≤ 2(1 +Rs)
3. (4.41)
By (4.40), Q0v is given by Q0v = T̂nTnQ0u and the P -equation (4.12) becomes
Q(λ)u = PnQ0u+ PnT̂nTnQ0u.
Since Id + T̂nTn = T̂n one is led to 0 =
(
Q(λ) − PnT̂nQ0
)
u. Hence given any
|n| ≥ 2C0R
2
s(q, ϕ), λ ∈ Πn is a periodic eigenvalue of Q iff det(Sn(λ)) = 0, where
Sn(λ) ≡ Sn(λ, q, ϕ) is the map
Sn(λ) =
(
Q(λ)− PnT̂nQ0
)
Pn : Pn → Pn. (4.42)
496 Thomas Kappeler and Yannick Widmer
We now compute the matrix representation of Sn with respect to the basis
[e
(1)
−n, e
(2)
n ] of Pn. By (4.9), the matrix representation [Q(λ)] of Q(λ) is given
by
[Q(λ)] =
(
λ− nπ
λ− nπ
)
and for any |n| ≥ 2C0R
2
s(q, ϕ), the one of PnT̂nQ0Pn is given by(
a+
n (λ) b+n (λ)
b−n (λ) a−n (λ)
)
:=
〈T̂nQ0e
(1)
−n, e
(1)
−n
〉
c
〈
T̂nQ0e
(2)
n , e
(1)
−n
〉
c〈
T̂nQ0e
(1)
−n, e
(2)
n
〉
c
〈
T̂nQ0e
(2)
n , e
(2)
n
〉
c
. (4.43)
For any ρ ≥ 1, denote by B̃s+1
ρ the closed ball of radius ρ in H̃s+1
c , centered at 0,
B̃s+1
ρ :=
{
(q, ϕ) ∈ H̃s+1
c : 1 +Rs(q, ϕ) ≤ ρ
}
, (4.44)
where we recall that Rs(q, ϕ) = ‖ϕ‖s + ‖cosh(q/2)‖s+1 + ‖sinh(q/2)‖s+1.
Lemma 4.7. Let s ≥ 0, ρ ≥ 1, and |n| ≥ 2C0ρ
4. Then the following holds:
(i) A complex number λ ∈ Πn is a periodic eigenvalue of Q1∂x +Q0 iff
detSn(λ) = (λ− πn− an(λ))2 − b+n (λ)b−n (λ)
vanishes.
(ii) The functions a±n , b±n are analytic in (λ, (q, ϕ)) on Πn × B̃s+1
ρ . Furthermore
an := a+
n coincides with a−n and
an(λ, q, ϕ) = an(λ, q,−ϕ) , b−n (λ, q, ϕ) = b+n (λ, q,−ϕ) .
Proof. (i) The statement follows from the definition of Sn as mentioned in
the discussion above.
(ii) By Lemma 4.5 (i), T 4
n with |n| ≥ 2C0ρ
4 is a 1
2 -contraction for any element
in Πn×B̃s+1
ρ . Hence (Id−T 4
n)−1 can be expanded in its Neumann series, implying
that an(λ) and b±n (λ) can be written as series which converge normally and are
analytic on Πn × B̃s+1
ρ . Note that detQ0(x) = (detB(x))2 = 1
4 . Hence Q0(x) is
invertible for any x and T̂nQ0 is invertible as an operator. By the definition of
Tn = Q0Q(λ)−1Kn it then follows that(
Q−1
0 (Id− Tn)
)∗
=
(
Id− (Q(λ)−1Kn)∗Q∗0
)
(Q∗0)−1
= (Q∗0)−1
(
Id−Q∗0(Q(λ)−1Kn)∗
)
.
Using that the adjoints of Q0, Q(λ)−1Kn with respect to 〈·, ·〉c are given by
Q0(q, ϕ)∗ = Q0(q,−ϕ) and (Q(λ)−1Kn)∗ = KnQ(λ)−1 = Q(λ)−1Kn
one has(
Q−1
0 (q, ϕ)(Id− Tn(λ, q, ϕ))
)∗
=
(
Q−1
0 (q,−ϕ)(Id− Tn(λ, q,−ϕ))
)
.
On Spectral Properties of the L Operator. . . 497
Taking the inverse of both sides of the latter identity one gets(
T̂n(λ, q, ϕ)Q0(q, ϕ)
)∗
= T̂n
(
λ, q,−ϕ
)
Q0(q,−ϕ). (4.45)
Hence b+n (λ, q, ϕ) = b−n (λ, q,−ϕ) and a±n (λ, q, ϕ) = a±n (λ, q,−ϕ). It remains to
prove that a+
n = a−n . For a given linear operator B acting on a C-vector space,
denote by B its complex conjugate defined by Bu := Bu. Furthermore, note that(
Z
Z
)
e(1)
n = e(2)
n and
(
Z
Z
)
e(2)
n = e(1)
n .
Using that e
(2)
−n = e
(2)
n and 〈a, b〉c = 〈b, a〉c one then gets
a+
n =
〈
T̂nQ0e
(1)
−n, e
(1)
−n
〉
c
=
〈
T̂nQ0
(
Z
Z
)
e
(2)
−n,
(
Z
Z
)
e
(2)
−n
〉
c
=
〈
e
(2)
−n,
(
Z
Z
)
(T̂nQ0)∗
(
Z
Z
)
e
(2)
−n
〉
c
=
〈(
Z
Z
)
(T̂nQ0)∗
(
Z
Z
)
e(2)
n , e(2)
n
〉
c
.
It remains to compute (
Z
Z
)
(T̂nQ0)∗
(
Z
Z
)
.
A straightforward computation yields
Q0(q, ϕ)∗ =
(
Z
Z
)
Q0(q, ϕ)
(
Z
Z
)
,
(Q(λ)−1Kn)∗ =
(
Z
Z
)
Q(λ)−1Kn
(
Z
Z
)
.
Hence the adjoint of T̂nQ0 =
(
Id−Q0Q(λ)−1Kn
)−1Q0 is given by
(T̂nQ0)∗ =
(
Z
Z
)
(T̂nQ0)
(
Z
Z
)
.
This implies that a+
n (λ) = a−n (λ).
For a function f : U → C, defined on a domain U ⊂ X of a C-Banach space
(X, ‖·‖), denote by |f |U its sup norm,
|f |U := sup
λ∈U
‖f(λ)‖ .
498 Thomas Kappeler and Yannick Widmer
Lemma 4.8. Let (q, ϕ) ∈ H̃1
c and |n| ≥ 2C0R
4
0(q, ϕ). Then
|an|Πn ≤
1
|n|
(1 +R0(q, ϕ))4R2
0(q, ϕ) + (1 +R0(q, ϕ))4 ‖ϕ‖L2 R0;|n|(ϕ). (4.46)
Furthermore, if in addition ϕ ∈ Hs
C for some s ≥ 0, one has R0;|n|(ϕ) ≤ 1
〈n〉s ‖ϕ‖s
and hence
|an|Πn ≤ 2(1 +R0(q, ϕ))4
(
R2
0(q, ϕ)
|n|
+
‖ϕ‖L2 ‖ϕ‖s
〈n〉s
)
. (4.47)
Proof. Since T̂n = Id+ T̂nTn = Id+ Tn + T̂nT
2
n and
Q0e
(1)
−n =
1
4
(0, −ϕe−n, cosh(q/2)e−n, − sinh(q/2)e−n), (4.48)
one has 〈Q0e
(1)
−n, e
(1)
−n〉c = 0. Using that T̂n = Id + Tn + T̂nT
2
n we split an into a
sum, an = Σ1 + Σ2, where
Σ1 :=
〈
TnQ0e
(1)
−n, e
(1)
−n
〉
c
, Σ2 :=
〈
T̂nT
2
nQ0e
(1)
−n, e
(1)
−n
〉
c
.
Substitute
ϕ =
∑
m∈Z
ϕmem where ϕm ≡ ϕ̂(m) =
∫ 1
0
ϕ(x)e−imπx dx
into TnQ0e
(1)
−n = Q0Q(λ)−1KnQ0e
(1)
−n to obtain
TnQ0e
(1)
−n = Q0Q(λ)−1Kn
1
4
0
−ϕe−n
cosh(q/2)e−n
− sinh(q/2)e−n
= Q0Q(λ)−1 1
4
0
−
∑
m6=n ϕm+nem
cosh(q/2)e−n
− sinh(q/2)e−n
and by (4.9)
Q(λ)−1 1
4
0
−
∑
m 6=n ϕm+nem
cosh(q/2)e−n
− sinh(q/2)e−n
=
1
4
0
−
∑
m6=n ϕm+n
em
λ−nπ
1
λ cosh(q/2)e−n
− 1
λ sinh(q/2)e−n
.
Using trigonometric identities, one then concludes
TnQ0e
(1)
−n =
1
16λ
cosh(q)e−n
− sinh(q)e−n
0
0
− 1
16
∑
m6=n
ϕm+n
λ−mπ
em
ϕ
0
− sinh(q/2)
cosh(q/2)
. (4.49)
On Spectral Properties of the L Operator. . . 499
Hence by (4.19), for any λ ∈ Πn,
|Σ1| =
∣∣∣〈TnQ0e
(1)
−n, e
(1)
−n
〉
c
∣∣∣
≤ 1
16 |λ|
∣∣ ̂cosh(q)(0)
∣∣+
1
16
∑
m6=n
|ϕm+n|
|n−m|
|ϕ−(m+n)|. (4.50)
Since cosh(q) = cosh2(q/2) + sinh2(q/2) one has by Lemma 4.1
‖cosh(q)‖L2 ≤2 ‖cosh(q/2)‖21 + 2 ‖sinh(q/2)‖21 (4.51)
and hence∣∣ ̂cosh(q)(0)
∣∣ ≤ ‖cosh(q)‖L2 ≤ 2 ‖cosh(q/2)‖21 + 2 ‖sinh(q/2)‖21 .
For the second term in (4.50), we split the sum into two parts, |m−n| > |n| and
1 ≤ |m− n| ≤ |n| to get
1
16
∑
m6=n
|ϕm+n|
|n−m|
|ϕ−(m+n)| ≤
1
16 |n|
‖ϕ‖2L2 +
1
16
∑
1≤|m−n|≤|n|
|ϕm+n||ϕ−(m+n)| .
Using that for |m − n| ≤ |n| one has |m + n| = |2n + m − n| ≥ 2|n| − |n| = |n|
and hence ∑
1≤|m−n|≤|n|
|ϕm+n||ϕ−(m+n)| ≤ ‖ϕ‖L2 R0;|n|(ϕ).
Altogether we thus have shown that
|Σ1| ≤
R2
0
8|n|
+
1
16
‖ϕ‖L2 R0;|n|(ϕ).
Towards Σ2 note that for any vector valued L2-function f and |〈f, e(i)
−n〉c| ≤
‖f‖L2 , 1 ≤ i ≤ 4. Hence for f = T̂nT
2
nQ0e
(1)
−n,
|Σ2| =
∣∣∣〈T̂nT 2
nQ0e
(1)
−n, e
(1)
−n
〉
c
∣∣∣ ≤ ∥∥∥T̂nT 2
nQ0e
(1)
−n
∥∥∥
L2
.
Hence by (4.41)
|Σ2| ≤
∥∥∥T̂nT 2
nQ0e
(1)
−n
∥∥∥
L2
≤ 2 (1 +R0)3
∥∥∥T 2
nQ0e
(1)
−n
∥∥∥
L2
. (4.52)
Furthermore, by (4.49),
T 2
nQ0e
(1)
−n =
1
16λ
Tn
cosh(q)e−n
− sinh(q)e−n
0
0
− 1
64λ
∑
m 6=n
ϕm+n
λ−mπ
em
− sinh(q)
cosh(q)
0
0
− Tn
1
16
∑
m6=n
ϕm+n
λ−mπ
ϕe(1)
m . (4.53)
500 Thomas Kappeler and Yannick Widmer
We will now estimate the three terms in the latter expression separately. One
easily checks, using Lemma 4.2, that for any λ ∈ Πn∥∥∥∥∥∥∥∥
1
16λ
Tn
cosh(q)e−n
− sinh(q)e−n
0
0
∥∥∥∥∥∥∥∥
L2
≤ 1
16|n|
R0 (‖cosh(q)‖L2 + ‖sinh(q)‖L2) .
For the second term in (4.53) one has for any λ ∈ Πn∥∥∥∥∥∥∥∥
1
64λ
∑
m 6=n
ϕm+n
λ−mπ
em
− sinh(q)
cosh(q)
0
0
∥∥∥∥∥∥∥∥
L2
≤ 1
64|n|
(‖sinh(q)‖L2 + ‖cosh(q)‖L2)‖ϕ‖L2 .
Finally, for the last term in (4.53) one has by Lemma 4.2∥∥∥∥∥∥Tn 1
16
∑
m 6=n
ϕm+n
λ−mπ
ϕe(1)
m
∥∥∥∥∥∥
L2
≤ R0
1
16
‖ϕ‖L2
∑
m6=n
|ϕm+n|2
|n−m|2
1/2
,
where by arguing as above,∑
m6=n
|ϕm+n|2
|n−m|2
1/2
≤ 1
|n|
‖ϕ‖L2 +R0;|n|(ϕ).
Altogether we have proved∥∥∥T 2
nQ0e
(1)
−n
∥∥∥
L2
≤ R0
4 |n|
(
‖ϕ‖2L2 +
1
2
(‖cosh(q)‖L2 + ‖sinh(q)‖L2)
)
+
R0
16
‖ϕ‖L2 R0;|n|(ϕ).
By Lemma 4.1 ‖sinh(q)‖L2 ≤ 4 ‖cosh(q/2)‖1 ‖sinh(q/2)‖1 and with (4.51) one
obtains
‖ϕ‖2L2 +
1
2
(‖cosh(q)‖L2 + ‖sinh(q)‖L2)
≤ ‖ϕ‖2L2 + (‖cosh(q/2)‖1 + ‖sinh(q/2)‖1)2 ≤ R2
0.
Hence by (4.52),
|Σ2| ≤
1
2|n|
(1 +R0)3R3
0 + (1 +R0)3R0 ‖ϕ‖L2 R0;|n|(ϕ).
Combining the estimates for Σ1 and Σ2 yields (4.46). The estimate (4.47) imme-
diately follows from (4.46).
Next we estimate b+n (λ), b−n (λ), introduced in (4.43).
On Spectral Properties of the L Operator. . . 501
Lemma 4.9. Let s ≥ 0, (q, ϕ) ∈ H̃s+1
c and λ ∈ Πn with |n| ≥ 2C0R
4
s. Then
the following holds:
〈n〉s
∣∣∣∣b±n (λ)∓ 1
4
ϕ̂(∓2n)
∣∣∣∣
Πn
≤ 1
2|n|
(1 +Rs(q, ϕ))6 , (4.54)
|b+n b−n |Πn ≤
|ϕ̂(−2n)|2 + |ϕ̂(2n)|2
16
+
1
2n2
(1 +R0(q, ϕ))12 , (4.55)
〈n〉2s|b+n b−n |Πn ≤
1
16
‖ϕ‖2s +
1
2n2
(
1 +Rs(q, ϕ)
)12
. (4.56)
Furthermore ∑
n≥2C0(1+Rs)4+1
〈n〉2s6|b+n b−n |Πn
1/2
≤ ‖ϕ‖s + 2
(
1 +Rs(q, ϕ)
)4
. (4.57)
Proof. We begin by proving the estimate (4.54) for b−n (λ) for n with |n| ≥
2C0R
4
s. By the definition (4.43), b−n (λ) = 〈T̂nQ0e
(1)
−n, e
(2)
n 〉c. Arguing as in the
proof of Lemma 4.8, one gets for any λ ∈ Πn
b−n (λ) =
〈
Q0e
(1)
−n, e
(2)
n
〉
c
+
〈
TnQ0e
(1)
−n, e
(2)
n
〉
c
+
〈
T̂nT
2
nQ0e
(1)
−n, e
(2)
n
〉
c
.
By (4.48), one has〈
Q0e
(1)
−n, e
(2)
n
〉
c
= −1
4
∫ 1
0
ϕe−ne−n dx = −1
4
ϕ̂(2n).
By (4.49) and (4.6), for any λ ∈ Πn∣∣∣〈TnQ0e
(1)
−n, e
(2)
n 〉c
∣∣∣ =
1
16|λ|
∣∣ ̂sinh(q)(2n)
∣∣ ≤ 1
16|λ|〈2n〉s
‖sinh(q)‖s
≤ 1
4|n|〈2n〉s
‖sinh(q/2)‖s+1 ‖cosh(q/2)‖s+1 .
By (4.41),
〈n〉s
∣∣∣〈T̂nT 2
nQ0e
(1)
−n, e
(2)
n
〉
c
∣∣∣ ≤ ∥∥∥T̂nT 2
nQ0e
(1)
−n
∥∥∥
s;n
≤ 2(1 +Rs)
3
∥∥∥T 2
nQ0e
(1)
−n
∥∥∥
s;n
.
Since by (4.48), the first entry of Q0e
(1)
−n vanishes one can apply Lemma 4.5 (iii)
to obtain
∥∥∥T 2
nQ0e
(1)
−n
∥∥∥
s;n
≤ 1
|n|R
2
s
∥∥∥Q0e
(1)
−n
∥∥∥
s;n
. Similarly, by the formula (4.48) for
Q0e
(1)
−n one has
∥∥∥Q0e
(1)
−n
∥∥∥
s;n
≤ 1
4Rs and the claimed estimate of b−n (λ) of (4.54)
follows. The estimate for b+n (λ) is proved in a similar fashion. To prove (4.55)
and (4.56) use that |ab| ≤ 1
2(|a|2 + |b|2) to obtain for λ ∈ Πn
|b+n b−n | ≤
(∣∣∣∣b+n − 1
4
ϕ̂(−2n)
∣∣∣∣+
∣∣∣∣14 ϕ̂(−2n)
∣∣∣∣)(∣∣∣∣b−n +
1
4
ϕ̂(2n)
∣∣∣∣+
∣∣∣∣14 ϕ̂(2n)
∣∣∣∣)
502 Thomas Kappeler and Yannick Widmer
≤
∣∣∣∣b+n − 1
4
ϕ̂(−2n)
∣∣∣∣2 +
∣∣∣∣14 ϕ̂(−2n)
∣∣∣∣2 +
∣∣∣∣b−n +
1
4
ϕ̂(2n)
∣∣∣∣2 +
∣∣∣∣14 ϕ̂(2n)
∣∣∣∣2 .
Hence by (4.54), for any λ ∈ Πn,
|b+n b−n | ≤
1
16
(
|ϕ̂(−2n)|2 + |ϕ̂(2n)|2
)
+
1
2n2〈n〉2s
(1 +Rs)
12 . (4.58)
For s = 0, this yields (4.55) and for s ≥ 0 arbitrary (4.56). Finally since∑
n≥2C0(1+Rs)4+1
〈n〉2s|ϕ̂(−2n)|2 + 〈n〉2s|ϕ̂(2n)|2 ≤ ‖ϕ‖2s
and since C0 ≥ 1 and therefore∑
n≥2C0(1+Rs)4+1
1
n2
≤
∫ ∞
2C0(1+Rs)4
1
x2
dx =
1
2C0(1 +Rs)4
(4.57) follows from (4.58).
We now prove the following stronger version of Theorem 1.1.
Theorem 4.10. Let s ≥ 0 and (q, p) ∈ Hs+1
c . Then there exists N1 ≥ 2C0(1+
Rs)
4 + 1 so that for any |n| ≥ N1, detSn = (λ− πn− an(λ))2 − b+n (λ)b−n (λ) has,
when counted with multiplicities, exactly two roots λ±n in Πn. They are contained
in Dn ⊂ Πn. Furthermore, γn = λ+
n − λ−n satisfy
|γn|2 ≤ 6|b+n b−n |Πn , n ≥ N1 (4.59)
and ∑
n≥N1
〈n〉2s|γn(q, p)|2
1/2
≤‖ϕ(q, p)‖s + 2
(
1 +Rs(q, p)
)4
(4.60)
where by a slight abuse of terminology,
Rs(q, p) ≡ Rs(q, ϕ(q, p)) = ‖ϕ‖s + ‖sinh(q/2)‖s+1 + ‖cosh(q/2)‖s+1 .
Remark 4.11. Recall that by the reciprocity law, for any n ≥ 0,
1
16λ−−n(q, p)
− 1
16λ+
−n(q, p)
= λ+
n (−q, p)− λ−n (−q, p) = γn(−q, p).
Hence (4.59) applied to (−q, p) leads to the estimate∣∣∣∣∣ 1
16λ−−n(q, p)
− 1
16λ+
−n(q, p)
∣∣∣∣∣
2
≤ 6|b+n b−n |Πn,−q,p, n ≥ N1.
On Spectral Properties of the L Operator. . . 503
Proof. By assumption (q, ϕ) ∈ H̃s+1. According to Lemma 4.8 and 4.9, there
exists N1 ≥ 2C0(1 + R0)4 + 1 so that for any |n| ≥ N1, |an|Πn , |b+n b−n |Πn ≤ π
48 .
Hence, for any |n| ≥ N1 and λ ∈ Πn∣∣detSn(λ)− (λ− nπ − an(λ))2
∣∣ ≤ |b+n (λ)b−n (λ)|2 ≤
( π
48
)2
and
inf
λ∈∂Dn
|λ− nπ − an(λ)|2 ≥
∣∣∣∣π3 − sup
λ∈∂Dn
|an(λ)|
∣∣∣∣2 > ( π48
)2
.
As detSn(λ) and (λ−nπ−an(λ))2 are both analytic on Πn, by Rouché’s Theorem,
they have the same number of roots in Dn when counted with multiplicities. By
the same argument, one shows that (λ−nπ−an(λ))2 and (λ−nπ)2 have the same
number of roots in Dn when counted with multiplicities. Hence detSn(λ) has two
roots in Dn. By choosing N1 larger than the integer N in Lemma 3.11 (Counting
Lemma), it follows that these two roots are precisely the periodic eigenvalues λ+
n
and λ−n .
To prove the claimed estimate for the gaps γn = λ+
n − λ−n , we write
detSn = (λ− πn− an)2 − b+n b−n = g+g−, (4.61)
where
g± = λ− nπ − an ± σn, σn =
√
b+n b
−
n , (4.62)
and the choice of the branch of the root does not matter. Each root ξn ∈ Dn of
detSn is either a root of g+ or g− and hence is of the form ξn = nπ + an(ξn) ±
σ(ξn). It then follows that
|λ+
n−λ−n | ≤ |an(λ+
n )−an(λ−n )|+|σn(λ+
n )|+|σn(λ−n )| ≤ |∂λan|Dn |λ+
n−λ−n |+2|σn|Πn .
Since dist(Dn, ∂Πn) ≥ π/6, we obtain from Cauchy’s estimate |∂λan|Dn ≤
|an|Πn
π/6 ≤ 1
8 , which implies that 7
8 |λ
+
n − λ−n | ≤ 2
∣∣∣√b+n b−n ∣∣∣
Πn
and therefore |λ+
n −
λ−n |2 ≤ 6|b+n b−n |Πn . By Lemma 4.9 one then gets that∑
n≥N1
〈n〉2s|γn|2
1/2
≤ ‖ϕ‖s + 2 (1 +Rs)
4 .
Together with Lemma 2.14 (reciprocity in λ), this yields (4.59) and (4.60).
Remark 4.12. Assume that (q, p) ∈ H1
c is real valued. Then −ϕ = ϕ and hence
by Lemma 4.7
b−n (λ, q, ϕ) = b+n (λ, q, ϕ), an(λ, q, ϕ) ∈ R, ∀λ ∈ R∗.
Furthermore, by Lemma 4.9 (s = 0),
|b+n (λ±n )− 1
4
ϕ̂(−2n)| = O
(
1
n
)
504 Thomas Kappeler and Yannick Widmer
and hence
σn(λ±n ) = |b+n (λ±n )| = 1
4
|ϕ̂(−2n)|+O
(
1
n
)
.
On the other hand, by definition
an(λ) =
〈
(Id− Tn(λ))−1Q0e
(1)
−n, e
(1)
−n
〉
c
.
Expanding (Id− Tn(λ))−1 in the form
(Id− Tn(λ))−1 = Id+
3∑
k=1
Tn(λ)k + Tn(λ)4(Id− Tn(λ))−1
(4.5) and (4.48) yield
an(λ±n ) =
3∑
k=1
〈
Tn(λ±n )kQ0e
(1)
−n, e
(1)
−n
〉
c
+O
(
1
n
)
.
By (4.61), (4.62), there exists ρ±n ∈ {1,−1} so that λ±n = nπ+an(λ±n )+ρ±n σn(λ±n ).
It implies that
λ+
n−λ−n =
3∑
k=1
〈(Tn(λ+
n )k−Tn(λ−n )k)Q0e
(1)
−n, e
(1)
−n〉c+(ρ+
n−ρ−n )
1
4
|ϕ̂(−2n)|+O
(
1
n
)
.
For |n| sufficiently large the 2×2 matrix Sn contains all the information about
the nth periodic eigenvalues of a potential. In order to study their asymptotics
for |n| → ∞ in terms of the regularity of the potential, we analyze Sn(λ), λ ∈ Πn,
further. We will prove that the diagonal of Sn(λ) vanishes at a unique point λ =
σn(q, ϕ). These values will be used to locally define a real analytic perturbation
of the Fourier transform which allows to characterize the regularity of potentials
mentioned above. First we need to establish some auxiliary results.
Lemma 4.13. Let s > 0 and ρ > 0. Then for any (q, ϕ) ∈ B̃s+1
ρ and |n| ≥
max(2C0ρ
4, s
√
96ρ6, 96ρ6), there is a unique analytic function σn : B̃s+1
ρ → C
such that
(i) σn(q, ϕ) = nπ + an(σn(q, ϕ), (q, ϕ)), (q, ϕ) ∈ B̃s+1
ρ ,
(ii) sup
(q,ϕ)∈B̃s+1
ρ
|σn(q, ϕ)− nπ| ≤ π
48
,
(iii)σn(q, ϕ) ∈ R for any real valued (q, ϕ) ∈ B̃s+1
ρ .
Proof. For any given |n| ≥ max(2C0ρ
4, s
√
48ρ6, 96ρ6), consider the map T
with domain of definition E :=
{
σ : B̃s+1
ρ → D′n : σ real analytic
}
and D′n =
{λ ∈ C : |λ− nπ| ≤ π/48} ⊂ Dn, defined by
Tσ := nπ + an(σ(·), ·) .
On Spectral Properties of the L Operator. . . 505
The set E is obviously not empty since the constant function σ ≡ nπ is in E.
Note that by the definition of B̃s+1
ρ , the assumed lower bound for |n|, and Lemma
4.8,
|an|Πn×B̃s+1
ρ
≤ 2(1 +R0)4
(
R2
0
|n|
+
‖ϕ‖L2 ‖ϕ‖s
〈n〉s
)
≤ 2ρ4
(
ρ2
96ρ6
+
ρ2
96ρ6
)
≤ π
48
implying that T maps E into E. Endow E with the metric d(σ1, σ2) = |σ1 −
σ2|B̃s+1
ρ
. Then E is complete. We claim that T is a contraction. Indeed
d(T (σ1), T (σ2)) = |T (σ1)− T (σ2)|B̃s+1
ρ
≤ |∂λan|D′n×B̃s+1
ρ
d(σ1, σ2) ≤ 1
23
d(σ1, σ2)
as by Cauchy’s estimate
|∂λan|D′n×B̃s+1
ρ
≤
|an|Πn×B̃s+1
ρ
dist(D′n, ∂Πn)
≤ π/48
π/2− π/48
=
1
23
.
Hence T admits a unique fixed point in E, denoted by σn. By construction, σn
satisfies (i), (ii) and item (iii) holds since, by the uniqueness of σn and Lemma
4.7 (ii), one has σn(q,−ϕ) = σn(q, ϕ).
Let s > 0 and ρ > 0. Then for any |n| ≥ max(2C0ρ
4, s
√
96ρ6) and (q, ϕ) ∈
B̃s+1
ρ ,
Sn(σn(q, ϕ), q, ϕ) =
(
0 −b+n (σn(q, ϕ), q, ϕ)
−b−n (σn(q, ϕ), q, ϕ) 0
)
.
By Lemma 4.9, we know that b+n (λ) is close to 1
4 ϕ̂(−2n) and b−n (λ) is close to
−1
4 ϕ̂(2n). For any given s > 0, define the perturbed Fourier series Fs,ρ(q, ϕ) ∈
Hs
C for (q, ϕ) ∈ B̃s+1
ρ as follows:
Fs,ρ(q, ϕ) :=
∑
|n|≤Ms,ρ+1
ϕnen
+
∑
n>Ms,ρ+1
4b+n (σn(q, ϕ), q, ϕ)e−2n − 4b−n (σn(q, ϕ), q, ϕ)e2n, (4.63)
where
Ms,ρ := max(2C0ρ
4, s
√
96ρ6, 96ρ6, 220+2sρ10). (4.64)
The choice of Ms,ρ ensures that Fs,ρ is a local diffeomorphism (see proof of Lemma
4.14 below). Furthermore we introduce
Φs,ρ : B̃s+1
ρ → H̃s+1
c , (q, ϕ) 7→ (q,Fs,ρ(q, ϕ)).
Lemma 4.14. Let ρ ≥ 1 and s > 0. Then Φs,ρ : B̃s+1
ρ → Φs
ρ(B̃
s+1
ρ ) ⊂ H̃s+1
c
is a real analytic diffeomorphism such that
‖ϕ‖s
2
≤ ‖Fs,ρ(q, ϕ)‖s ≤ 2 ‖ϕ‖s , (q, ϕ) ∈ B̃s+1
ρ ,
506 Thomas Kappeler and Yannick Widmer
implying that B̃s+1
ρ/2 ⊂ Φs,ρ(B̃
s+1
ρ ). Moreover,
sup
(q,ϕ)∈B̃sρ
∥∥∂ϕFs,ρ(q, ϕ)− IdHs
C
∥∥
s
≤ 1
4
.
Proof. For any |n| ≥Ms,ρ ≥ 2C0ρ
4, σn maps B̃s+1
2ρ into Πn (cf. Lemma 4.13
(ii)) and b±n (σn(q, ϕ), q, ϕ) is well defined for (q, ϕ) ∈ B̃s+1
2ρ . By (4.54),
〈n〉s
∣∣4b+n (σn(q, ϕ), q, ϕ)− ϕ̂(−2n)
∣∣
B̃s2ρ
≤ 〈n〉s
∣∣4b+n − ϕ̂(−2n)
∣∣
Πn×B̃s2ρ
≤ 2
|n|
(2ρ)6, (4.65)
〈n〉s
∣∣4b−n (σn(q, ϕ), q, ϕ) + ϕ̂(2n)
∣∣
B̃s2ρ
≤ 〈n〉s
∣∣4b−n + ϕ̂(2n)
∣∣
Πn×B̃s2ρ
≤ 2
|n|
(2ρ)6. (4.66)
Hence the map Fs,ρ is defined on B̃s+1
2ρ and takes values in Hs(T,C). Moreover,
by the definition of Fs,ρ and (4.65), (4.66),
sup
(q,ϕ)∈B̃s+1
2ρ
‖Fs,ρ(q, ϕ)− ϕ‖2s ≤
∑
n>Ms,ρ+1
〈2n〉2s
∣∣4b+n (σ(q, ϕ), q, ϕ)− ϕ̂(−2n)
∣∣2
B̃s+1
2ρ
+ 〈2n〉2s
∣∣4b−n (σ(q, ϕ), q, ϕ) + ϕ̂(2n)
∣∣2
B̃s+1
2ρ
≤ 22s
∑
n>Ms,ρ+1
8
n2
(2ρ)12 ≤ 216+2s ρ
12
Ms,ρ
≤ ρ2
16
.
By Cauchy’s estimate applied to Fs,ρ(q, ·) on B̃s+1
ρ
sup
(q,ϕ)∈B̃s+1
ρ
∥∥∂ϕFs,ρ(q, ϕ)− IdHs
C
∥∥
s
≤ 1
ρ
sup
(q,ϕ)∈B̃s+1
2ρ
‖Fs,ρ(q, ϕ)− ϕ‖s ≤
1
4
.
Hence dϕFs,ρ(q, ϕ) : Hs(T,C)→ Hs(T,C) is invertible for any (q, ϕ) ∈ B̃s+1
ρ and
so is
dΦs,ρ =
(
Id 0
∂qFs,ρ ∂ϕFs,ρ
)
.
We thus have proved that for any s > 0, the map Φs,ρ : B̃s+1
ρ → H̃s+1
c is a local
diffeomorphism. Furthermore,∣∣‖Fs,ρ(q, ϕ)‖s − ‖ϕ‖s
∣∣ ≤ ‖Fs,ρ(q, ϕ)− ϕ‖s
≤ sup
(q,ϕ)∈B̃s+1
ρ
∥∥∂ϕFs,ρ(q, ϕ)− IdHs(T,C)
∥∥
s
‖ϕ‖ < 1
4
‖ϕ‖ .
Hence
‖ϕ‖s
2
≤ ‖Fs,ρ(q, ϕ)‖s ≤ 2 ‖ϕ‖s , ∀(q, ϕ) ∈ B̃s+1
ρ .
On Spectral Properties of the L Operator. . . 507
To see that Φs,ρ : B̃s+1
ρ → H̃s+1
c is one-to-one, note that for (q, ϕ1), (q, ϕ2) ∈
B̃s+1
ρ
‖Fs,ρ(q, ϕ1)−Fs,ρ(q, ϕ2)− (ϕ1 − ϕ2)‖s
≤ sup
(q,ϕ)∈B̃s+1
ρ
∥∥∂ϕFs,ρ − IdHs(T,C)
∥∥ ‖ϕ1 − ϕ2‖s ≤
1
4
‖ϕ1 − ϕ2‖s .
Thus if Fs,ρ(q, ϕ1) = Fs,ρ(q, ϕ2), one has ‖ϕ1 − ϕ2‖s ≤
1
4 ‖ϕ1 − ϕ2‖ which implies
ϕ1 = ϕ2.
Finally, we come to the proof of Theorem 1.2. As already mentioned, we
actually prove a slightly stronger version of this theorem, stated below. Recall
that a potential (q, p) ∈ H1
c is said to be a right [left] sided N -gap potential with
N ∈ Z≥0 if
∀n > N γn(q, p) = 0 [∀n > N γ−n(q, p) = 0]. (4.67)
It is said to be a right [left] sided finite gap potential if it is a right [left] sided
N -gap potential for some N ∈ Z. Denote by LFGsc and RFGsc the following
subsets of Hs
c :
LFGsc := {(q, p) ∈ Hs
c : (q, p) left sided finite gap potential}
and
RFGsc := {(q, p) ∈ Hs
c : (q, p) right sided finite gap potential} .
Theorem 4.15.
(i) For any s ∈ R≥1, LFGsc and RFGsc are dense in Hs
c .
(ii) For any s ∈ R≥1, the sets LFGsc ∩Hs
r and RFGsc ∩Hs
r are dense in the real
Hilbert space Hs
r := {v ∈ Hs
c : v real valued}.
Proof. (i) Since H̃s+1
c and Hs+1
c are isomorphic (cf. (4.7)) and H̃s+1
c is dense
in H̃s
c , it suffices to prove that, for any s ∈ R≥1 and ρ ≥ 1, the sets
Ls+1
ρ :=
{
(q, ϕ) ∈ B̃s+1
ρ : (q, ϕ) left sided finite gap potential
}
and
Rs+1
ρ :=
{
(q, ϕ) ∈ B̃s+1
ρ : (q, ϕ) right sided finite gap potential
}
are dense in B̃s+1
ρ . By a slight abuse of terminology, we say that (q, ϕ) = (q, Pp+
qx) is a right or left sided finite gap potential if (q, p) is such a potential. Let
us first prove that Rs+1
ρ is dense in B̃s+1
ρ . For any M ∈ Z≥1, denote by Gs,M
the closed subspace of Hs(T,C) spanned by e2k = ei2kπx, |k| ≤ M . Then Gs,M
is an increasing sequence of subspaces of Hs(T,C) and
⋃
M≥M0
Gs,M is dense
in Hs(T,C). Hence
⋃
M≥M0
Hs+1(T,C) × Gs,M is dense in H̃s+1
c . Here M0 =
max(Ms,ρ, N1), where Ms,ρ is given by (4.63) and N1 by Theorem 4.10. Since,
by Lemma 4.14, Φs,ρ : B̃s+1
ρ → Φs,ρ(B̃
s+1
ρ ) is a (real analytic) diffeomorphism, it
508 Thomas Kappeler and Yannick Widmer
follows that the preimage of Φs,ρ(B̃
s+1
ρ ) ∩
(⋃
M≥M0
Hs+1(T,C)× Gs,M
)
is dense
in B̃s+1
ρ . We claim that any element in this set is a right sided finite gap po-
tential. Indeed, if for any given (q, ϕ) ∈ B̃s+1
ρ , Φs,ρ(q, ϕ) = (q,Fs,ρ(q, ϕ)) is
in Hs+1(T,C) × Gs,M for some M ≥ M0, then by the definition (4.63) of Fs,ρ,
b−n (σn(q, ϕ), q, ϕ) = 0 and b+n (σn(q, ϕ), q, ϕ) = 0 for any n > M . Since (q, ϕ) ∈
B̃s+1
ρ and M ≥ Ms,ρ. Sn(λ, q, ϕ) is well defined for λ ∈ Πn (cf. (4.42)). Since
M ≥ N1, it follows from Theorem 4.10 that
detSn(σn(q, ϕ)) =
(
σn(q, ϕ)− πn− an(σn(q, ϕ))
)2 − b+n b−n ∣∣∣
σn(q,ϕ)
= 0.
Note that σn(q, ϕ) is a double root of
(
σn(q, ϕ)− πn− an(σn(q, ϕ))
)2
as well as
a double root of b+n b
−
n hence it is a double root of detSn in Πn, implying that
γn(q, ϕ) = 0. It means that (q, ϕ) is a right sided M -gap potential. We thus have
shown that Rs+1
ρ is dense in B̃s+1
ρ . Using Lemma 2.14 (reciprocity in λ), one
sees that the arguments above applied to (−q, ϕ) yield that Ls+1
ρ is also dense in
B̃s+1
ρ . This proves (i). Item (ii) is proved in the same way.
Supports. Both authors were supported by the Swiss National Science
Foundation.
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On Spectral Properties of the L Operator. . . 509
Received February 15, 2018.
Thomas Kappeler,
Institut für Mathematik, Universität Zürich, Winterthurerstrasse 190, CH-8057 Zürich,
Switzerland,
E-mail: thomas.kappeler@math.uzh.ch
Yannick Widmer,
Institut für Mathematik, Universität Zürich, Winterthurerstrasse 190, CH-8057 Zürich,
Switzerland,
E-mail: Yannick.Widmer@math.uzh.ch
Про спектральнi властивостi оператора L
у парi Лакса рiвняння sine-Гордона
Thomas Kappeler and Yannick Widmer
Ми аналiзуємо перiодичний спектр оператора L у парi Лакса рiвнян-
ня sine-Гордона в термiнах регулярностi потенцiалу.
Ключовi слова: рiвняння sine-Гордона, рiвняння sinh-Гордона, пара
Лакса, властивостi спадання довжин лакун.
mailto:thomas.kappeler@math.uzh.ch
mailto:Yannick.Widmer@math.uzh.ch
Introduction
Fundamental solution
Spectra
Proofs of Theorem 1.1 and Theorem 1.2
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