Spectral Gaps of the One-Dimensional Schrödinger Operators with Singular Periodic Potentials
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irk-123456789-57022010-02-03T12:01:03Z Spectral Gaps of the One-Dimensional Schrödinger Operators with Singular Periodic Potentials Mikhailets, V. Molyboga, V. 2009 Article Spectral gaps of the one-dimensional Schrodinger operators with singular periodic potentials / V. Mikhailets, V. Molyboga // Methods of Functional Analysis and Topology. — 2009. — Т. 15, № 1. — С. 31-40. — Бібліогр.: 31 назв. — англ. 1029-3531 http://dspace.nbuv.gov.ua/handle/123456789/5702 en Інститут математики НАН України |
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Mikhailets, V. Molyboga, V. Spectral Gaps of the One-Dimensional Schrödinger Operators with Singular Periodic Potentials |
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Mikhailets, V. Molyboga, V. |
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Mikhailets, V. |
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Spectral Gaps of the One-Dimensional Schrödinger Operators with Singular Periodic Potentials |
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Spectral Gaps of the One-Dimensional Schrödinger Operators with Singular Periodic Potentials |
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Spectral Gaps of the One-Dimensional Schrödinger Operators with Singular Periodic Potentials |
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Spectral Gaps of the One-Dimensional Schrödinger Operators with Singular Periodic Potentials |
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Spectral Gaps of the One-Dimensional Schrödinger Operators with Singular Periodic Potentials |
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spectral gaps of the one-dimensional schrödinger operators with singular periodic potentials |
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Інститут математики НАН України |
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2009 |
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Spectral gaps of the one-dimensional Schrodinger operators with singular periodic potentials / V. Mikhailets, V. Molyboga // Methods of Functional Analysis and Topology. — 2009. — Т. 15, № 1. — С. 31-40. — Бібліогр.: 31 назв. — англ. |
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2025-07-02T08:46:57Z |
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2025-07-02T08:46:57Z |
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1836524260059774976 |
| fulltext |
Methods of Functional Analysis and Topology
Vol. 15 (2009), no. 1, pp. 31–40
SPECTRAL GAPS OF THE ONE-DIMENSIONAL SCHRÖDINGER
OPERATORS WITH SINGULAR PERIODIC POTENTIALS
VLADIMIR MIKHAILETS AND VOLODYMYR MOLYBOGA
To the memory of A. Ya. Povzner
Abstract. The behavior of the lengths of spectral gaps {γn(q)}∞
n=1 of the Hill-
Schrödinger operators
S(q)u = −u′′ + q(x)u, u ∈ Dom (S(q)) ,
with real-valued 1-periodic distributional potentials q(x) ∈ H−1
1-per(�) is studied. We
show that they exhibit the same behavior as the Fourier coefficients {�q(n)}∞
n=−∞
of
the potentials q(x) with respect to the weighted sequence spaces hs,ϕ, s > −1,
ϕ ∈ SV. The case q(x) ∈ L2
1-per(�), s ∈ �+, ϕ ≡ 1, corresponds to the Marchenko-
Ostrovskii Theorem.
1. Introduction
The Hill-Schrödinger operators
S(q)u := −u′′ + q(x)u, u ∈ Dom(S(q))
with real-valued 1-periodic distributional potentials q(x) ∈ H−1
1-per(R) are well defined on
the Hilbert space L2(R) in the following equivalent basic ways [21]:
• as form-sum operators;
• as quasi-differential operators;
• as limits of operators with smooth 1-periodic potentials in the norm resolvent
sense.
The operators S(q) are lower semibounded and self-adjoint on the Hilbert space L2(R).
Their spectra are absolutely continuous and have a band and gap structure as in the
classical case of L2
1-per(R)-potentials [9, 13, 4, 21].
The object of our investigation is the behavior of the lengths of spectral gaps. Under
the assumption that
(1) q(x) =
∑
k∈Z
q̂(k)eik2πx ∈ H−1+
1-per(R, R),
that is, ∑
k∈Z
(1 + 2|k|)2s|q̂(k)|2 < ∞ ∀s > −1, and Im q(x) = 0,
we will prove many terms asymptotic estimates for the lengths {γn(q)}∞n=1 and midpoints
{τn(q)}∞n=1 of spectral gaps of the Hill-Schrödinger operators S(q) (Theorem 1). These
estimates enable us to establish a relationship between the rate of decreasing/increasing
of the lengths of the spectral gaps and the regularity of the singular potentials (Theorem 2
and Theorem 3).
2000 Mathematics Subject Classification. Primary 34L40; Secondary 47A10, 47A75.
Key words and phrases. Hill–Schrödinger operators, singular potentials, spectral gaps.
31
32 VLADIMIR MIKHAILETS AND VOLODYMYR MOLYBOGA
It is well known that if the potentials satisfy
q(x) =
∑
k∈Z
q̂(k)eik2πx ∈ L2
1-per(R, R), Im q(x) = 0,
i.e., if ∑
k∈Z
|q̂(k)|2 < ∞ and q̂(k) = q̂(−k) ∀k ∈ Z,
then the Hill-Schrödinger operators S(q) are lower semibounded and self-adjoint on the
Hilbert space L2(R) with absolutely continuous spectra that have a zone structure [5, 28].
The spectra spec (S(q)) are defined by the location of the endpoints {λ0(q), λ
±
n (q)}∞n=1
of the spectral gaps, and the endpoints satisfy the following inequalities:
−∞ < λ0(q) < λ−
1 (q) ≤ λ+
1 (q) < λ−
2 (q) ≤ λ+
2 (q) < · · · .
Moreover, for even/odd numbers n ∈ Z+, the endpoints of the spectral gaps are eigen-
values of the periodic/semiperiodic problems on the interval [0, 1],
S±(q)u := −u′′ + q(x)u = λu,
Dom(S±(q)) :=
{
u ∈ H2[0, 1]
∣∣∣u(j)(0) = ± u(j)(1), j = 0, 1
}
≡ H2
±[0, 1].
The spectral bands (stability or tied zones),
B0(q) := [λ0(q), λ
−
1 (q)], Bn(q) := [λ+
n (q), λ−
n+1(q)], n ∈ N,
are characterized as a locus of those real λ ∈ R for which all solutions of the equation
S(q)u = λu are bounded. On the other hand, the spectral gaps (instability or forbidden
zones),
G0(q) := (−∞, λ0(q)), Gn(q) := (λ−
n (q), λ+
n (q)), n ∈ N,
are a locus of those real λ ∈ R for which any nontrivial solution of the equation S(q)u =
λu is unbounded.
Due to Marchenko and Ostrovskii [14], the endpoints of spectral gaps of the Hill-
Schrödinger operators S(q) satisfy the asymptotic estimates
(2) λ±
n (q) = n2π2 + q̂(0) ± |q̂(n)| + h1(n), n → ∞.
As a consequence, for the lengths of spectral gaps,
γn(q) := λ+
n − λ−
n , n ∈ N,
the following asymptotic formulae hold:
(3) γn(q) = 2 |q̂(n)| + h1(n), n → ∞.
Hochstadt [10] (⇒) and Marchenko, Ostrovskii [14], McKean, Trubowitz [15] (⇐)
proved that the potential q(x) is an infinitely differentiable function if and only if the
lengths of spectral gaps {γn(q)}∞n=1 decrease faster than an arbitrary power of 1/n,
q(x) ∈ C∞
1-per(R, R) ⇔ γn(q) = O(n−k), n → ∞ ∀k ∈ Z+.
Marchenko and Ostrovskii [14] discovered that
(4) q(x) ∈ Hk
1-per(R, R) ⇔ {γn(q)}∞n=1 ∈ hk, k ∈ Z+.
The relationship (4) was extended by Kappeler, Mityagin [11] (⇒) and Djakov, Mitya-
gin [2] (⇐) (see also the survey [3] and the references therein) to the case of symmetric,
monotone, submultiplicative and subexponential weights Ω = {Ω(n)}n∈Z
,
q(x) ∈ HΩ
1-per(R, R) ⇔ {γn(q)}∞n=1 ∈ hΩ.
Pöschel [27] proved the latter statement in a quite different way.
SPECTRAL GAPS OF THE ONE-DIMENSIONAL SCHRÖDINGER OPERATORS 33
Here and throughout the rest of the paper we use the complex Hilbert spaces Hw
1-per(R)
(as well as Hw
± [0, 1]) of 1-periodic functions and distributions defined by means of their
Fourier coefficients
f(x) =
∑
k∈Z
f̂(k)eik2πx ∈ Hw
1-per(R) ⇔
{
f̂(k)
}
k∈Z
∈ hw,
hw =
{
a = {a(k)}k∈Z
∣∣∣∣ ‖a‖hw =
( ∑
k∈Z
w2(k)|a(k)|2
)1/2
< ∞
}
.
Basically we use the power weights
ws = {ws(k)}k∈Z
: ws(k) := (1 + 2|k|)s, s ∈ R.
The corresponding spaces we denote by
Hws
1-per(R) ≡ Hs
1-per(R), Hws
± [0, 1] ≡ Hs
±[0, 1], and hws ≡ hs, s ∈ R.
For more details, see Appendix.
2. Main results
As we have already remarked, if assumption (1) is satisfied, the Hill-Schrödinger op-
erators S(q) are lower semibounded and self-adjoint on the Hilbert space L2(R). Their
spectra are absolutely continuous and have a classical zone structure [9, 13, 4, 21, 23].
Using the results of the papers [12, 26], the Isospectral Theorem 5, and [21, Theorem C]
we obtain uniform many terms asymptotic estimates for the lengths of spectral gaps,
{γn(q)}∞n=1, and for their midpoints {τn(q)}∞n=1,
τn(q) :=
λ+
n (q) + λ−
n (q)
2
, n ∈ N.
Theorem 1. ([18, 25]). Let q(x) ∈ H−α
1-per(R, R), α ∈ [0, 1). Then for any ε > 0,
uniformly on bounded sets of distributions q(x) in the corresponding Sobolev spaces
H−α
1-per(R), the lengths {γn(q)}∞n=1 and the midpoints {τn(q)}∞n=1 of spectral gaps of the
Hill-Schrödinger operators S(q) for n ≥ n0
(
‖q‖H−α
1-per
(R)
)
satisfy the following asymp-
totic formulae:
γn(q) = 2 |q̂(n)| + h1−2α−ε(n),(5)
τn(q) = n2π2 + q̂(0) + h1−2α−ε(n).(6)
Corollary. ([18, 25]). Let q(x) ∈ H−α
1-per(R, R) with α ∈ [0, 1). Then for any ε > 0,
uniformly in q(x), for the endpoints of spectral gaps of the Hill-Schrödinger operators
S(q) the following asymptotic estimates hold:
λ±
n (q) = n2π2 + q̂(0) ± |q̂(n)| + h1−2α−ε(n).
Now, we can describe a two-way relationship between the rate of decreasing/increasing
of the lengths of spectral gaps, {γn(q)}∞n=1, and regularity of the potentials q(x) in a
refined scale.
Let
ws,ϕ = {ws,ϕ(k)}k∈Z
: ws,ϕ(k) := (1 + 2|k|)s ϕ(|k|), s ∈ R, ϕ ∈ SV,
where ϕ is a function slowly varying at +∞ in the sense of Karamata [30]. This means
that it is a function that is positive, measurable on [a,∞), a > 0, and obeys the condition
lim
t→+∞
ϕ(λt)
ϕ(t)
= 1, λ > 0.
34 VLADIMIR MIKHAILETS AND VOLODYMYR MOLYBOGA
For example,
ϕ(t) = (log t)r1(log log t)r2 . . . (log . . . log t)rk ∈ SV, {r1, . . . , rk} ⊂ R, k ∈ N.
The Hörmander spaces
H
ws,ϕ
1-per(R) ≡ Hs,ϕ
1-per(R) ≃ Hs,ϕ(S), S := R/2πZ,
and the weighted sequence spaces
hws,ϕ ≡ hs,ϕ
form the refined scales:
Hs+ε
1-per(R) →֒ Hs,ϕ
1-per(R) →֒ Hs−ε
1-per(R),(7)
hs+ε →֒ hs,ϕ →֒ hs−ε, s ∈ R, ε > 0, ϕ ∈ SV,(8)
which, in a general situation, were studied by Mikhailets and Murach [22].
The following statements show that the sequence {γn(q)}∞n=1 has the same behavior
as the Fourier coefficients {q̂(n)}∞n=−∞ with respect to the refined scale {hs,ϕ}s∈R,ϕ∈SV.
Theorem 2. Let q(x) ∈ H−1+
1-per(R, R). Then
q(x) ∈ Hs,ϕ
1-per (R, R) ⇔ {γn(q)}∞n=1 ∈ hs,ϕ, s ∈ (−1, 0], ϕ ∈ SV.
Note that the Hörmander spaces Hs,ϕ
1-per(R) with ϕ ≡ 1 coincide with the Sobolev
spaces,
Hs,1
1-per(R) ≡ Hs
1-per(R), and hs,1 ≡ hs, s ∈ R.
Corollary. ([18, 25]). Let q(x) ∈ H−1+
1-per(R, R), then
(9) q(x) ∈ Hs
1-per (R, R) ⇔ {γn(q)}∞n=1 ∈ hs, s ∈ (−1, 0].
Theorem 2, together with [11, Theorem 1.2], and properties (7) and (8), gives the
following extension of the Marchenko-Ostrovskii Theorem (4).
Theorem 3. Let q(x) ∈ H−1+
1-per(R, R). Then
q(x) ∈ Hs,ϕ
1-per (R, R) ⇔ {γn(q)}∞n=1 ∈ hs,ϕ, s ∈ (−1,∞), ϕ ∈ SV.
In particular,
q(x) ∈ Hs
1-per (R, R) ⇔ {γn(q)}∞n=1 ∈ hs, s ∈ (−1,∞).
Remark. In the preprint [4], the authors have announced, without a proof, a more general
statement,
q(x) ∈ H
�Ω
1-per(R, R) ⇔ {γn(q)}∞n=1 ∈ h
�Ω, Ω̂ =
{
Ω(n)
1 + 2|n|
}
n∈Z
,
where the weights Ω = {Ω(n)}n∈Z are supposed to be symmetric, monotone, submulti-
plicative and subexponential. This result contains the limiting case
q(x) ∈ H−1
1-per (R, R) \ H−1+
1-per (R, R) .
The implication
q(x) ∈ H−1
1-per (R, R) ⇒ {γn(q)}∞n=1 ∈ h−1
was proved in the paper [13].
SPECTRAL GAPS OF THE ONE-DIMENSIONAL SCHRÖDINGER OPERATORS 35
3. Proofs
Spectra of the Hill-Schrödinger operators S(q), q(x) ∈ H−1
1-per (R, R) are defined by the
endpoints {λ0(q), λ
±
n (q)}∞n=1 of spectral gaps. The endpoints as in the case of L2
1-per(R)-
potentials satisfy the inequalities
−∞ < λ0(q) < λ−
1 (q) ≤ λ+
1 (q) < λ−
2 (q) ≤ λ+
2 (q) < · · · .
For even/odd numbers n ∈ Z+ they are eigenvalues of the periodic/semiperiodic problems
on the interval [0, 1] [21, Theorem C],
S±(q)u = λu.
The operators
S±u ≡ S±(q)u := D2
±u + q(x)u,
• D2
± := −d2/dx2, Dom(D2
±) = H2
±[0, 1];
• q(x) =
∑
k∈Z
q̂(k) ei k2πx ∈ H−1
+ ([0, 1], R) ;
• Dom(S±(q)) =
{
u ∈ H1
±[0, 1]
∣∣ D2
±u + q(x)u ∈ L2(0, 1)
}
,
are well defined on the Hilbert space L2(0, 1) as lower semibounded, self-adjoint form-sum
operators, and they have the pure discrete spectra
spec (S±(q)) =
{
λ0[S+(q)], λ±
2n−1[S−(q)], λ±
2n[S+(q)]
}∞
n=1
.
In the papers [18, 25, 19, 20] the authors meticulously investigated the more general
periodic/semiperiodic form-sum operators
Sm,±(V ) := D2m
± ∔ V (x), V (x) ∈ H−m
+ [0, 1], m ∈ N,
on the Hilbert space L2(0, 1).
So, we need to find precise asymptotic estimates for eigenvalues of the operators S±(q).
It is quite a difficult problem for the form-sum operators S±(q) are not convenient for
studying. We also cannot apply the approach developed by Savchuk and Shkalikov
(see the survey [29] and the references therein) considering the operators S±(q) as quasi-
differential, since the periodic/semiperiodic boundary conditions are not strongly regular
in the sense of Birkhoff. Therefore, we propose an alternative approach which is based
on isospectral transformation of the problem.
Kappeler and Möhr [12, 26] investigated the second order differential operators L±(q),
q(x) ∈ H−1
+ ([0, 1], R) (in general, with complex-valued potentials) defined on the negative
Sobolev spaces H−1
± [0, 1],
L± ≡ L±(q) := D2
± + q(x), Dom(L±(q)) = H1
±[0, 1].
They established that the operators L±(q) with q(x) ∈ H−α
+ ([0, 1], R), α ∈ [0, 1), have
the real-valued discrete spectra
spec (L±(q)) =
{
λ0[L+(q)], λ±
2n−1[L−(q)], λ±
2n[L+(q)]
}∞
n=1
such that ∣∣λ±
n [L±(q)] − n2π2 − q̂(0)
∣∣ ≤ Cnα, n ≥ n0
(
‖q‖H−α
+
[0,1]
)
.
More precisely, for the values
γn[L±(q)] := λ+
n [L±(q)] − λ−
n [L±(q)], n ∈ N,
τn[L±(q)] :=
λ+
n [L±(q)] + λ−
n [L±(q)]
2
, n ∈ N,
they proved the following result.
36 VLADIMIR MIKHAILETS AND VOLODYMYR MOLYBOGA
Proposition 4. (Kappeler, Möhr [12, 26]). Let q(x) ∈ H−α
+ ([0, 1], R), and α ∈ [0, 1).
Then for any ε > 0, uniformly on bounded sets of distributions q(x) in the Sobolev spaces
H−α
+ [0, 1], the values {γn[L±(q)]}
∞
n=1 and {τn[L±(q)]}
∞
n=1, n ≥ n0
(
‖q‖H−α
+
[0,1]
)
, for the
operators L±(q) satisfy the following asymptotic estimates:
i)
{
min
±
∣∣∣γn[L±(q)] ± 2
√
(q̂ + ω )(−n)( q̂ + ω) (n)
∣∣∣
}
n∈N
∈ h1−2α−ε,
ii) τn[L±(q)] = n2π2 + q̂(0) + h1−2α−ε(n),
where
{ω(n)}n∈Z
≡
{
1
π2
∑
k∈Z\{±n}
q̂ (n − k)q̂(n + k)
n2 − k2
}
n∈Z
∈
{
h1−α, α ∈ [0, 1/2),
h3/2−2α−δ, α ∈ [1/2, 1)
with any δ > 0 (see the Convolution Lemma [12, 26]).
Remark. In the papers [24, 16, 17, 25], the more general operators
Lm,±(V ) := D2m
± + V (x), V (x) ∈ H−m
+ [0, 1], m ∈ N,
on the spaces H−m
± [0, 1] were studied. In particular, an analogue of Proposition 4 was
proved.
The following statement is an essential point of our approach.
Theorem 5. (Isospectral Theorem [18, 25]). The operators S±(q) and L±(q) are isospec-
tral,
spec (S±(q)) = spec (L±(q)) .
Proof. The inclusions
spec (S±(q)) ⊂ spec (L±(q))
are obvious, since
S±(q) ⊂ L±(q).
Let us prove the inverse inclusions,
spec (L±(q)) ⊂ spec (S±(q)) .
Let λ ∈ spec (L±(q)), and f be a correspondent eigenvector or a rootvector. Therefore
(L±(q) − λId) f = g, f, g ∈ Dom(L±(q)) = H1
±[0, 1],
where f is an eigenfunction if g = 0, and a rootvector if g �= 0.
So, we get
L±(q)f = λIdf + g ∈ H1
±[0, 1],
i.e.,
L±(q)f = D2
±f + q(x)f ∈ L2(0, 1).
Thus we have proved that f ∈ Dom(S±(q)). In the case when f is a rootvector (g �= 0)
in a similar fashion we show that g ∈ Dom(S±(q)), too. Continuing this process as
necessary (note that it is finite, since the eigenvalue λ has finite algebraic multiplicity)
we obtain that all eigenvectors and rootvectors corresponding to λ belong to the domains
Dom (S±(q)) of the operators S±(q). Consequently, we can conclude that
λ ∈ spec (S±(q)) ,
hence we obtain the needed inclusions,
spec (L±(q)) ⊂ spec (S±(q)) .
The proof is complete. �
SPECTRAL GAPS OF THE ONE-DIMENSIONAL SCHRÖDINGER OPERATORS 37
Now, Theorem 1 follows from Proposition 4, the Isospectral Theorem 5, and [21,
Theorem C], since
q̂(n) = q̂(−n), n ∈ Z,
ω(n) = ω(−n), n ∈ Z,
and, as a consequence,
min
±
∣∣∣γn(q) ± 2
√
(q̂ + ω )(−n)( q̂ + ω) (n)
∣∣∣ = |γn(q) − 2 |(q̂ + ω) (n)|| .
The proof of Theorem 1 is complete.
To prove Theorem 2 we firstly prove its Corollary. The formula (9) follows from [12,
Corollary 0.2 (2.6)], the Isospectral Theorem 5 and [21, Theorem C]. Also it can be
proved directly as well similarly to [12, Corollary 0.2 (2.6)] using estimates (5).
Now, to prove Theorem 2 it is sufficient to apply the asymptotic estimates (5), prop-
erties (7) and (8) of the refined scales, and formula (9),
• q ∈ Hs,ϕ
1-per (R, R)
(7)
=⇒ q ∈ Hs−δ
1-per (R, R) , δ > 0
(5)
=⇒ γn = 2 |q̂(n)| + h1+2(s−δ)−ε(n)
(8)
=⇒ γn = 2 |q̂(n)| + hs,ϕ(n) =⇒ {γn(q)}∞n=1 ∈ hs,ϕ;
• {γn(q)}∞n=1 ∈ hs,ϕ (8)
=⇒ {γn}
∞
n=1 ∈ hs−δ, δ > 0
(9)
=⇒ q ∈ Hs−δ
1-per (R, R)
(5)
=⇒ γn = 2 |q̂(n)| + h1+2(s−δ)−ε(n)
(8)
=⇒ γn = 2 |q̂(n)| + hs,ϕ(n)
=⇒ {q̂(n)}n∈Z ∈ hs,ϕ(n).
Note that, since δ > 0 and ε > 0 were chosen arbitrarily, we can take them to be such
that
1 + s − 2δ − ε > 0.
The proof of Theorem 2 is complete.
Now, we are ready to prove Theorem 3.
At first, note that from [11, Theorem 1.2] we get the following asymptotic formulae
for the lengths of spectral gaps:
(10) γn(q) = 2 |q̂(n)| + h1+s(n) for q(x) ∈ Hs
1-per (R, R) , s ∈ [0,∞),
which, for integer numbers s ∈ Z+, were proved by Marchenko and Ostrovskii [14].
Using (9), (10) and (4) it is easy to prove that
(11) q(x) ∈ Hs
1-per (R, R) ⇔ {γn(q)}∞n=1 ∈ hs, s ∈ (−1,∞).
Sufficiency in Theorem 3. Let q(x) ∈ Hs,ϕ
1-per (R, R). If s ∈ (−1, 0], then using Theo-
rem 2 we obtain that {γn(q)}∞n=1 ∈ hs,ϕ. If s > 0, then
q(x) ∈ Hs,ϕ
1-per (R, R)
(7)
→֒ Hs−δ
1-per (R, R) , δ > 0
(10)
=⇒ γn(q) = 2 |q̂(n)| + h1+s−δ(n)
(8)
=⇒ γn(q) = 2 |q̂(n)| + hs,ϕ(n) =⇒ {γn(q)}∞n=1 ∈ hs,ϕ.
Sufficiency is proved.
Necessity in Theorem 3. Let us assume that {γn(q)}∞n=1 ∈ hs,ϕ. If s ∈ (−1, 0] then
from Theorem 2 it follows that q(x) ∈ Hs,ϕ
1-per (R, R). If s > 0, then
{γn(q)}∞n=1 ∈ hs,ϕ (8)
→֒ hs−δ, δ > 0
(11)
=⇒ q(x) ∈ Hs−δ
1-per (R, R)
(10)
=⇒ γn(q) = 2 |q̂(n)| + h1+s−δ(n)
(8)
=⇒ γn(q) = 2 |q̂(n)| + hs,ϕ(n) =⇒ q(x) ∈ Hs,ϕ
1-per (R, R) .
38 VLADIMIR MIKHAILETS AND VOLODYMYR MOLYBOGA
Necessity is proved.
The proof of Theorem 3 is complete.
4. Concluding remarks
In fact, we can prove the following result: if q(x) ∈ H−1+
1-per (R, R) and
(1 + 2|k|)s ≪ w(k) ≪ (1 + 2|k|)1+2s, s ∈ (−1, 0],
(1 + 2|k|)s ≪ w(k) ≪ (1 + 2|k|)1+s, s ∈ [0,∞),
then
q(x) ∈ Hw
1-per (R, R) ⇔ {γn(q)}∞n=1 ∈ hw.
This result is not covered by the theorems in the preprint [4], because it does not
require the weight function to be monotone and submultiplicative.
Appendix
The complex Sobolev spaces Hs
1-per(R), s ∈ R, of 1-periodic functions and distributions
on the real axis R are defined by means of their Fourier coefficients,
Hs
1-per(R) :=
{
f =
∑
k∈Z
f̂(k)eik2πx
∣∣∣ ‖ f ‖Hs
1-per
(R)< ∞
}
,
‖ f ‖Hs
1-per
(R) :=
( ∑
k∈Z
〈2k〉2s|f̂(k)|2
)1/2
, 〈k〉 := 1 + |k|,
f̂(k) := 〈f, eik2πx〉L2
1-per
(R), k ∈ Z.
By 〈·, ·〉L2
1-per
(R) we denote the sesqui-linear form that gives the pairing between the dual
spaces Hs
1-per(R) and H−s
1-per(R) with respect to L2
1-per(R), and which is an extension by
continuity of the L2
1-per(R)-inner product [1, 8],
〈f, g〉L2
1-per
(R) :=
∫ 1
0
f(x)g(x) dx =
∑
k∈Z
f̂(k)ĝ(k) ∀f, g ∈ L2
1-per(R).
It is useful to notice that
H0
1-per(R) ≡ L2
1-per(R).
By Hs+
1-per(R), s ∈ R, we denote the inductive limit of the Sobolev spaces Ht
1-per(R)
with t > s,
Hs+
1-per (R) :=
⋃
ε>0
Hs+ε
1-per (R) .
It is a topological space with the inductive topology.
In a similar fashion the Sobolev spaces Hs
±[0, 1], s ∈ R, of 1-periodic (1-semiperiodic)
functions and distributions over the interval [0, 1] are defined by
Hs
±[0, 1] :=
{
f =
∑
k∈Γ±
f̂
(
k
2
)
eikπx
∣∣∣ ‖ f ‖Hs
±
[0,1]< ∞
}
,
‖ f ‖Hs
±
[0,1] :=
( ∑
k∈Γ±
〈k〉2s
∣∣∣∣f̂
(
k
2
) ∣∣∣∣
2)1/2
, 〈k〉 = 1 + |k|,
f̂
(
k
2
)
:= 〈f(x), eikπx〉±, k ∈ Γ±.
SPECTRAL GAPS OF THE ONE-DIMENSIONAL SCHRÖDINGER OPERATORS 39
Here
Γ+ ≡ 2Z := {k ∈ Z | k ≡ 0 (mod 2)} ,
Γ− ≡ 2Z + 1 := {k ∈ Z | k ≡ 1 (mod 2)} ,
and 〈·, ·〉± are sesqui-linear forms that define the pairing between the dual spaces Hs
±[0, 1]
and H−s
± [0, 1] with respect to L2(0, 1); the sesqui-linear forms 〈·, ·〉± are extensions by
continuity of the L2(0, 1)-inner product [1, 8],
〈f, g〉± :=
∫ 1
0
f(x)g(x) dx =
∑
k∈Γ±
f̂
(
k
2
)
ĝ
(
k
2
)
∀f, g ∈ L2(0, 1).
It is obvious that
H0
+[0, 1] ≡ H0
−[0, 1] ≡ L2(0, 1).
We say that a 1-periodic function or a distribution f(x) is real-valued if Im f(x) = 0.
Let us recall that
Re f(x) :=
1
2
(f(x) + f(x)), Im f(x) :=
1
2i
(f(x) − f(x)),
(see, for an example, [31]). In terms of the Fourier coefficients, we have
Im f(x) = 0 ⇔ f̂(k) = f̂(−k), k ∈ Z.
Set
Hs
1-per(R, R) :=
{
f(x) ∈ Hs
1-per(R) | Im f(x) = 0
}
,
Hs+
1-per(R, R) :=
{
f(x) ∈ Hs+
1-per(R) | Im f(x) = 0
}
,
Hs
± ([0, 1], R) :=
{
f(x) ∈ Hs
±[0, 1] | Im f(x) = 0
}
.
Also we will need the Hilbert spaces
hs ≡ hs (Z, C) , s ∈ R,
of (two-sided) weighted sequences,
hs :=
{
a = {a(k)}k∈Z
∣∣∣∣ ‖a‖hs :=
( ∑
k∈Z
〈k〉2s|a(k)|2
)1/2
< ∞
}
, 〈k〉 = 1 + |k|.
Note that
h0 ≡ l2 (Z, C) ,
and
a = {a(k)}k∈Z
∈ hs, s ∈ R, ⇒ a(k) = o(|k|−s), k → ±∞.
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(Russian)
Institute of Mathematics, National Academy of Sciences of Ukraine, 3 Tereshchenkivs’ka,
Kyiv, 01601, Ukraine
E-mail address: mikhailets@imath.kiev.ua
Institute of Mathematics, National Academy of Sciences of Ukraine, 3 Tereshchenkivs’ka,
Kyiv, 01601, Ukraine
E-mail address: molyboga@imath.kiev.ua
Received 01/09/2008
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