Expansion in Eigenfunctions of Relations Generated by Pair of Operator Differential Expressions
For relations generated by a pair of operator symmetric differential expressions, a class of generalized resolvents is found. These resolvents are integro-differential operators. The expansion in eigenfunctions of these relations is obtained.
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| Cite this: | Expansion in eigenfunctions of relations generated by pair of operator differential expressions / V. Khrabustovskyi // Methods of Functional Analysis and Topology. — 2009. — Т. 15, № 2. — С. 137-151. — Бібліогр.: 29 назв. — англ. |
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irk-123456789-57102010-02-03T12:01:13Z Expansion in Eigenfunctions of Relations Generated by Pair of Operator Differential Expressions Khrabustovskyi, V. For relations generated by a pair of operator symmetric differential expressions, a class of generalized resolvents is found. These resolvents are integro-differential operators. The expansion in eigenfunctions of these relations is obtained. 2009 Article Expansion in eigenfunctions of relations generated by pair of operator differential expressions / V. Khrabustovskyi // Methods of Functional Analysis and Topology. — 2009. — Т. 15, № 2. — С. 137-151. — Бібліогр.: 29 назв. — англ. 1029-3531 http://dspace.nbuv.gov.ua/handle/123456789/5710 en Інститут математики НАН України |
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For relations generated by a pair of operator symmetric differential expressions, a class of generalized resolvents is found. These resolvents are integro-differential operators. The expansion in eigenfunctions of these relations is obtained. |
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Khrabustovskyi, V. |
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Khrabustovskyi, V. Expansion in Eigenfunctions of Relations Generated by Pair of Operator Differential Expressions |
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Khrabustovskyi, V. |
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Khrabustovskyi, V. |
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Expansion in Eigenfunctions of Relations Generated by Pair of Operator Differential Expressions |
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Expansion in Eigenfunctions of Relations Generated by Pair of Operator Differential Expressions |
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Expansion in Eigenfunctions of Relations Generated by Pair of Operator Differential Expressions |
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Expansion in Eigenfunctions of Relations Generated by Pair of Operator Differential Expressions |
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Expansion in Eigenfunctions of Relations Generated by Pair of Operator Differential Expressions |
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expansion in eigenfunctions of relations generated by pair of operator differential expressions |
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Інститут математики НАН України |
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2009 |
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Expansion in eigenfunctions of relations generated by pair of operator differential expressions / V. Khrabustovskyi // Methods of Functional Analysis and Topology. — 2009. — Т. 15, № 2. — С. 137-151. — Бібліогр.: 29 назв. — англ. |
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2025-07-02T08:47:21Z |
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Methods of Functional Analysis and Topology
Vol. 15 (2009), no. 2, pp. 137–151
EXPANSION IN EIGENFUNCTIONS OF RELATIONS GENERATED
BY PAIR OF OPERATOR DIFFERENTIAL EXPRESSIONS
VOLODYMYR KHRABUSTOVSKYI
Dedicated to blessed memory of Professor Alexander Povzner
Abstract. For relations generated by a pair of operator symmetric differential ex-
pressions, a class of generalized resolvents is found. These resolvents are integro-
differential operators. The expansion in eigenfunctions of these relations is obtained.
1. The operator differential equation
(1) l[y] − λm[y] = m [f ] t ∈ Ī, I = (a, b) ⊆ R1
is considered on finite or infinite intervals in the space of vector-functions with values
in a separable Hilbert space H, where l[y] and m[y] are symmetric operator differen-
tial expressions of order r and s respectively, where r + s > 0, s is even. Expression
m[y] is non-negative and such that an operator first-order system obtained from the ho-
mogeneous equation (1) by using quasi-derivatives contains a spectral parameter λ in
Nevanlinna’s manner.
In the paper, the equation (1) reduces in a special way to a symmetric first order system
containing the spectral parameter either in a linear way (r > s) or in a nonlinear way
(r ≤ s). Using this reduction and the characteristic operator of this system (see [19], [20])
we construct a class of the generalized resolvents of the minimal relation corresponding to
(1). These resolvents are integro-differential operators. From this the inversion formulas
and Parseval’s equality are obtained. For their proof we modify Strauss’s method [27]
concerning the case of the generalized resovents as s = 0 and m[y] ≡ y which are integral
operators (but not integro-differential operators) depending on λ in a more simple way
(see [1], [4], [5], [6], [16], [17], [18], [27]) comparing with the case s > 0.
The expansion formulas in the solutions of the homogeneous equation (1) were ob-
tained in various particular situations in a number of papers. For dimH = 1 in the
regular case, r > s, and for special l and m this was done in [7, 14]. For dimH < ∞,
m[y] ≡ w(t)y, 0 ≤ w(t) ∈ B (H), the expansion formulas were obtained in [1] for r = 1
and, for the general case, in [16]–[18] (see also [26] for the case r = 1). Then for
dimH < ∞ the existence of the expansion formulas was proved in [11] under the as-
sumption that the leading coefficient of the expression m[y] is nondegenerate, and the
minimal differential operator that corresponds to this expression is uniformly positive on
any finite interval (i.e. under these assumptions even the case m[y] ≡ w(t)y with degen-
erate weight w(t) is not covered). However in [11], as it was mentioned by the authors,
the Titchmarsh-Kodaira’s formula for an explicit calculation of the spectral matrix is not
obtained. Also [11] does not contain an explicit expression for the resolvent and the case
r = s is not considered.
Further in the paper, the boundary value problems for the equation (1) with boundary
conditions depending on the spectral parameter are considered. We show that for some
2000 Mathematics Subject Classification. Primary 34B05, 34B07, 34L10.
Key words and phrases. Relations generated by pair of operator differential expressions, characteristic
operator, generalized resolvent, eigenfunction expansion.
137
138 VOLODYMYR KHRABUSTOVSKYI
boundary conditions, solutions of these problems are generated by a generalized resolvent
if, in contrast to the case s = 0, the boundary conditions contain the derivatives of vector-
function f(t) that are taken on the ends of interval.
In the general case where dimH < ∞ we find the absolutely continuous part of the
spectral matrix on the axis when the coefficients of the equation (1) are periodic on the
semi-axes and also we find the spectral matrix on the semi-axis when the coefficients are
periodic. These formulas are obtained using the results that are obtained in [18], [19] for
s = 0.
Notice that it is not supposed in the paper that for r ≥ s the leading coefficient of the
expression m[y] (which set the metric) has an inverse in B (H).
Many questions that concern differential operators and relations in the space of vector-
functions are considered in the monographs [1, 2, 3, 13, 23, 24, 25, 26] containing an
extensive literature. The method of studying these operators and relations based on a
use of the abstract Weyl function was proposed in [9].
We denote by ( . ) and ‖ · ‖ the scalar product and the norm in various spaces with
special indexes if it is necessary.
Let an interval ∆ ⊆ R, f(t) (t ∈ ∆) be a function with values in some Banach space
B. The notation f(t) ∈ Cl (∆) , l = 0, 1, . . . (we omit the index l if l = 0) means
that, in any point of ∆, f(t) has a norm ‖ · ‖B continuous derivatives of order up to and
including l that are taken in the norm ‖ ‖B; if ∆ is either semi-open or closed interval
then on its ends belonging to ∆ the one-side continuous derivatives exist. The notation
f(t) ∈ Cl
0 (∆) means that f(t) ∈ Cl (∆) and f(t) = 0 in the neighbourhoods of the ends
of ∆.
2. We consider an operator differential equation in a separable Hilbert space H1,
(2)
i
2
(
(Q(t)x(t))
′
+ Q∗(t)x′(t)
)
− Hλ(t)x(t) = Wλ(t)F (t), t ∈ Ī,
where Q(t), [ℜQ(t)]
−1
, Hλ(t) ∈ B (H1) , Q(t) ∈ C1
(
Ī
)
; the operator function Hλ(t) =
H∗
λ̄
(t) is continuous in t and is Nevanlinna’s in λ. Namely, the following condition holds:
The set A ⊇ C\R1 exists, any of its points has a neighbourhood independent of t ∈ Ī,
in this neighbourhood Hλ(t) is analytic ∀t ∈ Ī; ∀λ ∈ A Hλ(t) ∈ C
(
Ī
)
; the weight
Wλ(t) = ℑHλ(t)/ℑλ ≥ 0 (ℑλ �= 0).
In view of [20] ∀µ ∈ A⋂
R: Wµ(t) = ∂Hµ(t)/∂µ is Bochner locally integrable in the
uniform operator topology.
For convenience of statements we suppose that 0 ∈ Ī and we denote ℜQ (0) = G.
Let Xλ(t) be the operator solution of the homogeneous equation (2) satisfying the
initial condition Xλ (0) = I, where I is an identity operator in H1.
For any α, β ∈ Ī, α ≤ β we denote
∆λ (α, β) =
∫ β
α
X∗
λ(t)Wλ(t)Xλ(t)dt,
N = {h ∈ H1 |h ∈ Ker∆λ (α, β) ∀α, β } ,
P is an orthogonal projection onto N⊥. N is independent of λ ∈ A [20].
For x(t) ∈ H1 or x(t) ∈ B (H1) we denote
U [x(t)] = ([ℜQ(t)] x(t), x(t)) or U [x(t)] = x∗(t) [ℜQ(t)] x(t),
respectively.
As in [20] we introduce the following.
Definition 1. An analytic operator-function M (λ) = M∗
(
λ̄
)
∈ B (H1) of non-real λ is
called a characteristic operator (c.o.) of the equation (2) on I (or, simply, c.o.), if for
ℑλ �= 0 and for any H1-valued vector-function F (t) ∈ L2
Wλ
(I) with compact support the
EXPANSION IN EIGENFUNCTIONS OF RELATIONS 139
corresponding solution xλ(t) of the equation (3) of the form
(3)
xλ (t, F ) = RλF
=
∫
I
Xλ(t)
{
M (λ) − 1
2
sgn (s − t) (iG)−1
}
X∗
λ̄ (s)Wλ (s)F (s) ds
satisfies the condition
(4) (ℑλ) lim
(α, β)↑I
(U [xλ (β, F )] − U [xλ (α, F )]) ≤ 0 (ℑλ �= 0) .
The properties of c.o. and sufficient condition (that are close to necessary condition)
of the c.o.’s existence are obtained in [19, 20].
We consider in the separable Hilbert space H the equation (1), where l[y] and m[y] are
symmetric differential expressions of orders r and s correspondingly (one of these orders
can be equal to zero), where s is even, with sufficiently smooth coefficients from B (H).
Namely, l[y] =
∑r
k=0 iklk[y], where l2j = Djpj(t)D
j , l2j−1 = 1
2Dj−1
{
Dqj(t) +
q∗j (t)D
}
Dj−1, pj(t) = p∗j (t), qj(t) ∈ B (H), pj(t), qj(t) ∈ Cj(Ī), D = d/dt; m[y] is
defined in a similar way with s instead of r and p̃j(t) = p̃∗j (t), q̃j(t) ∈ B (H) instead of
pj(t), qj(t).
We denote by p(t, λ) the coefficient at the highest-order derivative in the homogeneous
equation (1), i.e.
p(t, λ) =
pn(t), r = 2n > s,
pn(t) − λp̃n(t), r = s = 2n,
−λp̃n(t), s = 2n > r,
iℜqn+1(t), r = 2n + 1 > s.
It is supposed in the paper that for non-real λ, p−1 (t, λ) ∈ B (H) for any t ∈ Ī. We
note that in that case where r = s the leading coefficients of both expressions l[y] and
m[y] may not have inverses in B (H) (in particular simultaneously) for any t ∈ Ī.
Denote p = max {r, s} and by y[k] (t |L) we denote the quasi-derivatives [21] of the
vector-function y(t) that corresponds to the differential expression L.
Using the substitution
(5)
x(t) = x (t, λ)
=
(∑n−1
j=0 ⊕y(j)(t)
)
⊕∑n
j=1 ⊕y[p−j] (t, |l − λm) ,
if p = 2n is even,
(∑n−1
j=0 ⊕y(j)(t)
)
⊕
(∑n
j=1 ⊕y[p−j] (t, |l − λm)
)
⊕
(
−iy(n)(t)
)
,
if p = 2n + 1 > 1 is odd,
y(t), if p = 1,
for t and λ such that p−1 (t, λ) ∈ B (H) the equation
(6) l[y] − λm[y] = 0, t ∈ Ī
is reduced to a homogenous equation of type (2) in H1 = Hp. Under this substitution
for odd p = r > s we formally consider that s = r − 1 and if it is necessary we set some
leading coefficients in the expression m[y] to be equal to zero. Analogously for even p we
formally consider that r = s.
Then the quasi-derivatives in (5) are equal to
y[j] (t|l − λm) = y(j)(t), j = 0, . . . , [p/2]− 1,(7)
140 VOLODYMYR KHRABUSTOVSKYI
y[n] (t |l − λm ) =
{
p (t, λ) y(n) − i
2 (qn − λq̃n) y(n−1), p = 2n,
− i
2 (qn+1 − λq̃n+1) y(n), p = 2n + 1,
(8)
y[p−j] (t |l − λm ) = −Dy[p−j−1] (t |l − λm ) + (pj − λp̃j) y(j)
+
i
2
[(
q∗j+1 − λq̃∗j+1
)
y(j+1) − (qj − λq̃j) y(j−1)
]
,(9)
j = 0, . . . ,
[
p − 1
2
]
, q0 ≡ q̃0 ≡ 0.
With this, l[y] − λm[y] = y[p] (t |l − λm ).
In the homogeneous equation (2) obtained from equation (6) using substitution (5)
for even p = 2n,
Q(t) = iJ =
(
0 iIn
−iIn 0
)
,
Hλ(t) = ‖hα β (t, λ)‖2
α, β=1 , hα β ∈ B (Hn) ,
(10)
where In is an identity operator in B (Hn), h11 (t, λ) = h∗
11
(
t, λ̄
)
is a three-diagonal
operator matrix whose elements under the main diagonal are equal to
( i
2
(q1 − λq̃1) , . . . ,
i
2
(qn−1λq̃n−1)
)
,
the elements on the main diagonal are equal to
(
− (p0 − λp̃0) , . . . ,
− (pn−2 − λp̃n−2)
1
4
(q∗n − λq̃∗n) p−1 (t, λ) (qn − λq̃n) − (pn−1 − λp̃n−1)
)
,
the rest of the elements are equal to zero. h12 (t, λ) = h∗
21
(
t, λ̄
)
is the operator ma-
trix with identical operators I1 under the diagonal, the elements on the diagonal are
equal to
(
0, . . . , 0, − i
2 (q∗n − λq∗n) p−1 (t, λ)
)
, the rest of the elements are equal to zero.
h22 (t, λ) = diag
(
0, . . . , 0, p−1 (t, λ)
)
.
And for odd p = 2n + 1,
(11)
Q(t) =
0 iIn 0
−iIn 0 0
0 0 qn+1
, p > 1,
q1, p = 1,
Hλ(t) =
{
‖hα β (t, λ)‖2
α, β=1 , p > 1,
p0 − λp̃0, p = 1,
where B (Hn) ∋ h11 (t, λ) = h∗
11
(
t, λ̄
)
is a three-diagonal operator matrix whose ele-
ments under the diagonal are equal
(
i
2 (q1 − λq̃1) , . . . , i
2 (qn−1 − λq̃n−1)
)
, the elements
on the diagonal are equal to (− (p0 − λp̃0) , . . . , − (pn−1 − λp̃n−1)), the rest of the el-
ements are equal to zero. B
(
Hn+1, Hn
)
� h12 (t, λ) = h∗
21
(
t, λ̄
)
is an operator ma-
trix whose elements with indices j, j − 1 are equal to I1, j = 2, . . . , n, the element
with index n, n + 1 is equal 1
2 (q∗n − λq∗n), the rest of the elements are equal to zero.
B
(
Hn+1
)
∋ h22 (t, λ) = h∗
22
(
t, λ̄
)
is an operator matrix whose last row is equal to
(0, . . . , 0, −iI1, − (pn − λp̃n)), the rest of elements are equal to zero.
Therefore in the equation (2) with coefficients (10), (11), Hλ(t) depend on λ in a
nonlinear manner for r ≤ s, and in a linear manner for r > s,
(12) Hλ(t) = H0(t) + λH(t), H∗
0 (t) = H0(t).
EXPANSION IN EIGENFUNCTIONS OF RELATIONS 141
Similarly to the general equation (2), for the equation (2) with coefficients (10),(11),
the weight is
Wλ(t) =
ℑHλ(t)/ℑλ, if ℑλ �= 0,
∂Hλ(t)
∂λ
, if ℑλ = 0, p−1(t, λ) ∈ B(H).
(13)
Everywhere below, unless stated otherwise, we assume that in the equation (2) with
coefficients (10), (11), Wλ(t) ≥ 0 (ℑλ �= 0).
Moreover, tacitly we assume that the following condition holds:
∃λ0 ∈ C; α, β ∈ Ī, 0 ∈ [α, β], the number δ > 0 :
p−1 (t, λ0) ∈ B (H) , ∀t ∈ [α, β],
m [χα, βy (t, λ0) , χα, βy (t, λ0)] ≥ δ ‖x (0, λ0)‖2
.
(14)
For any solution y (t, λ0) of the equation (6) as λ = λ0, where
m [f(t), g(t)] =
∫
I
∑s
k=0 mk [f(t), g(t)] dt,
m2j [f(t), g(t)] =
(
p̃2j(t)f
(j)(t), g(j)(t)
)
,
m2j−1 [f(t), g(t)] = i
2
{(
q̃∗j (t)f (j)(t), g(j−1)(t)
)
−
(
q̃j(t)f
(j−1)(t), g(j)(t)
)}
,
(15)
χα, β is a characteristic function of the interval (α, β), x (t, λ) is defined by (5).
For sufficiently smooth vector-function f(t) we denote
Hp ∋ Fλ(t) =
(∑s/2
j=0 ⊕f (j)(t)
)
⊕ O ⊕ . . .
. . . ⊕ O, r = 2n, r = 2n + 1 > 1, s < 2n,(∑n−1
j=0 ⊕f (j)(t)
)
⊕ O ⊕ . . .⊕
⊕O ⊕
(
−if (n)(t)
)
, r = 2n + 1 > 1, s = 2n,
f(t), r = 1,
an analog of (5) for f(t), r ≤ s.
(16)
Lemma 1. Let the vector-function f(t) ∈ Cs
(
Ī
)
, Fλ̄(t) be defined by (16) with λ̄ instead
of λ, Wλ(t) be defined by (13), (10), (11). Then
(17) Wλ(t)Fλ̄(t)=
(∑s/2−1
j=0 ⊕
(
f [s−j] (t|m) +
(
f [s−j−1] (t|m)
)′))
⊕
f [s/2] (t|m) ⊕ O ⊕ . . . ⊕ O,
r = 2n + 1, r = 2n, 0 < s < 2n,(∑n−1
j=0 ⊕
(
f [s−j] (t|m) +
(
f [s−j−1] (t|m)
)′))
⊕O ⊕ . . . ⊕ O ⊕
(
− if [n] (t|m)
)
,
r = 2n + 1 > 1, s = 2n,
p̃0(t)f(t) ⊕ O ⊕ . . . ⊕ O, s = 0,(∑n−1
j=0 ⊕
(
f [s−j] (t|m) +
(
f [s−j−1] (t|m)
)′))
⊕ O ⊕ . . .
. . . ⊕ O + Hλ(t)
(
O ⊕ . . . ⊕ O ⊕ f [n] (t|m)
)
, r ≤ s = 2n,
for λ, t such that p−1 (t, λ) ∈ B (H).
Proof. The proof for r > s follows from (7)–(11), (16).
Let r ≤ s = 2n. Let ℑλ �= 0. Since
(18) Wλ(t)Fλ̄ =
1
2iℑλ
((Hλ(t)Fλ(t) − Hλ̄(t)Fλ̄) + Hλ(t) (Fλ̄(t) − Fλ(t))) ,
using (7)–(10) and the fact that
(19) Hλ(t)Fλ(t) = iJ (Fλ(t))
′
− col
{
f [s] (t|l − λm) , 0, . . . , 0
}
,
we obtain (17) since ℑλ �= 0.
142 VOLODYMYR KHRABUSTOVSKYI
For λ0 ∈ R, t ∈ Ī which imply that p−1 (t, λ0) ∈ B (H), formula (17) is proved by
passing to the limit for λ → λ0 + i0. The lemma is proved. �
As is seen from the proof, Lemma 1 remains true without assuming that Wλ(t) ≥
0 (ℑλ �= 0) and (14).
Denote
q =
{
s/2, r > s,
s, r ≤ s.
Lemma 2. Let the vector-functions f(t), g(t) ∈ Cq
(
Ī
)
, Wλ(t) be defined by (13), (10),
(11). Then
(20)
s∑
k=0
mk [f(t), g(t)] = (Wλ(t)Fλ(t), Gλ(t))HP
for λ, t such that p−1 (t, λ) ∈ B (H) (Fλ(t) is defined by (16), Gλ(t) is defined in a
similar way using g(t) ) and, therefore,
(21) m [χα, βf(t), χα, β g(t)] = (Fλ(t), Gλ(t))L2
Wλ
(α, β)
for λ, t such that p−1(t, λ) ∈ B(H) ∀t ∈ [α, β] ⊆ Ī.
Proof. For r > s, (20) follows from (7)–(11), (16). For r ≤ s, (20) can be proved using
(5), (10), (16), (17) and (19). Lemma is proved. �
Note that the proof shows that the formula (20) is valid without the fulfilment of the
conditions Wλ(t) ≥ 0 (ℑλ �= 0) and (14).
In view of Lemma 2, the left-hand side of (20) is nonnegative for g(t) = f(t) since
Wλ(t) ≥ 0 in the equation (2), (10), (11), and the condition (14) is equivalent that for
this equation
∃λ0 ∈ C, α, β ∈ Ī, 0 ∈ [α, β], the number δ > 0 :
p−1 (t, λ0) ∈ B (H) , ∀t ∈ [α, β], (∆λ0
(α, β) g, g) ≥ δ ‖g‖2
, g ∈ Hp.
(22)
Therefore, in view of [20], fulfilment of (14) implies its fulfilment with δ (λ) > 0 instead
of the δ for λ ∈ C such that p−1 (t, λ) ∈ B (H), ∀ t ∈ [α, β].
Example 1. Let l[y] be a symmetric 2 × 2 -matrix differential operation of the second
order with a leading coefficient diag (p(t), 0), where p(t) �= 0, and
m[y] = −
((
0 0
0 q(t)
)
y′
)′
+ P (t)y,
where q(t) > 0, the operator P (t) = P ∗(t) > 0. In this case, det p (t, λ) �= 0 (ℑλ �= 0),
Wλ(t) ≥ 0 (ℑλ �= 0) and (14) holds, although det p1(t) ≡ det p̃1(t) ≡ detWλ(t) ≡ 0.
Definition 2. Every characteristic operator of the equation (2), (10), (11) corresponding
to the equation (1) is said to be a characteristic operator of the equation (1) on I (or
simply c.o.).
Lemma 3. 10. We establish a correspondence between the vector-function f(t) ∈ Cs
(
Ī
)
and the vector-function Fλ̄(t) that is obtained from (16) with λ̄ instead of λ.
Then equation (1) is equivalent to equation (2) with coefficients (10), (11), weight
(13) and with F (t) = Fλ̄(t) for such λ and t that p−1 (t, λ) ∈ B (H). Namely, if y(t) is
EXPANSION IN EIGENFUNCTIONS OF RELATIONS 143
a solution of the equation (1), then
(23)
x(t) = x (t, λ, f)
=
(∑n−1
j=0 ⊕y(j)(t)
)
⊕
(∑n−1
j=1 ⊕
(
y[r−j] (t |l − λm )−
−f [s−j] (t |m )
))
⊕ y[n] (t |l − λm ) , r = 2n > s,(∑n−1
j=0 ⊕y(j)(t)
)
⊕
(∑n
j=1 ⊕
(
y[r−j] (t|l − λm)−
−f [s−j] (t|m)
))
⊕
(
−iy(n)(t)
)
, r = 2n + 1 > s, r > 1,
y(t), r = 1,(∑n−1
j=0 ⊕y(j)(t)
)
⊕
(∑n
j=1 ⊕
(
y[s−j] (t|l − λm) −
− f [s−j] (t|m)
))
, r ≤ s = 2n
(here f [k] (t|m) ≡ 0 as k ≤ 0)
is a solution of (2) with coefficients (10), (11), weight (13) and with F (t) = Fλ̄(t). Any
solution of the equation (2) with coefficients (10), (11), weight (13) and with such F (t)
is equal to (23), where y(t) is a solution of (1).
20. Let M (λ) be a c.o. of the equation (1), Hp-valued vector-function F (t) ∈ L2
Wλ
(I)
(in particular, one can set F (t) = Fλ̄(t), where f(t) ∈ Cq
(
Ī
)
, m [f(t), f(t)] < ∞). Then
the integral (3) converges strongly and
(24) ‖RλF (t)‖2
L2
Wλ
(I) ≤ ℑ (RλF, F )L2
Wλ
(I) /ℑλ (ℑλ �= 0).
If, additionally, F (t) has compact support, then the inequality (24) is valid without
the requirement (14).
Proof. 10 is verified using direct calculations taking into account (7)–(11) and Lemma 1.
20 is proved in [20] for the general equation (2) satisfying a condition of type (22)
as λ0 ∈ A, and therefore it is proved for the equation (2) with coefficients (10), (11),
weight (13). The statement 20 for F (t) with compact support is also proved in [20] for
the general equation (2) without a condition of the type (22). Lemma is proved. �
We notice that one can see from the proof that item 1◦ of Lemma 3 is valid without
the fulfilment of the condition Wλ(t) ≥ 0 (ℑλ �= 0) and (14).
One can deduce from Lemmas 1–3 the following.
Corollary 1. Let the vector-functions x(t), y(t) ∈ Cp ([α, β]) , f(t), g(t) ∈ Cs ([α, β]),
p−1(t, λ), p−1(t, µ) ∈ B(H) ∀t ∈ [α, β] ⊆ Ī and
l[y] − λm[y] = m [f ] , l [x] − µm [x] = m [g] .
Then Green’s formula is valid,
m [χα, βf(t), χα, βx(t)] − m [χα, βy(t), χα, βg(t)] + (λ − µ̄)m [χα, βy(t), χα, βx(t)]
= ([iℜQ(t)]x (t, λ, f) , y (t, µ, g))|βα ,
where x (t, λ, f) is defined by (23), y (t, µ, g) is defined in a similar way using x(t)
instead of y(t), g instead of f and quasi-derivatives that correspond to the expression
l [x] − µm [x], Q(t) is defined by (10), (11).
We consider pre-Hilbert spaces
◦
H and H of vector-functions y(t) ∈ Cs
0
(
Ī
)
and y(t) ∈
Cs
(
Ī
)
, m [y(t), y(t)] < ∞, correspondingly, with the scalar product
(25) (f(t), g(t))m = m [f(t), g(t)] ,
where m [f(t), g(t)] is defined by (15).
144 VOLODYMYR KHRABUSTOVSKYI
Definition 3. By
◦
L 2
m (I) and L2
m (I) we denote the completions of the spaces
◦
H and
H in the norms ‖ · ‖m =
√
( · , · )m correspondingly. By
◦
P we denote the orthogonal
projection in L2
m (I) onto
◦
L2
m (I).
We consider, in L2
m (I), the symmetric relation
(26) L′
0 =
{{
ỹ(t), g̃(t)
}
|ỹ(t)
L2
m(I)
= y(t), g̃(t)
L2
m(I)
= g(t),
y(t) ∈ Cp
0
(
Ī
)
, g(t) ∈ Cs
0
(
Ī
)
, l[y] = m [g]
}
.
Further we assume that L′
0 consists of pairs of the type
{
y(t), g(t)
}
. We denote L0 = L′
0.
In the following theorem the generalized resolvents Rλ =
∫∞
−∞
dEµ
µ−λ of the relation
L0 are constructed and corresponding generalized spectral families Eµ = Eµ−0 [10] are
found. In this theorem we denote Eα, β = 1
2 (Eβ+0 + Eβ − Eα+0 − Eα) , −∞ < α ≤ β <
∞.
Theorem 1. 10. Let M (λ) be the characteristic operator of the equation (1),
(27) xλ (t, Fλ̄) = col {yj (t, λ, f)}p
j=1 (yj ∈ H) ,
be the corresponding solution (3) of the equation (2) with coefficients (10), (11), weight
(13) and F (t) = Fλ̄(t), where Fλ̄(t) is defined by (16) with λ̄ instead of λ, f(t) ∈ Cs
(
Ī
)
,
m [f(t), f(t)] < ∞ (and therefore Fλ̄(t) ∈ L2
Wλ
(I) in view of (20)).
Then an integro-differential operator Rλf = y1 (t, λ, f) which is densely defined in
L2
m (I) and given by the first vector-valued component of the solution (27) is, after closing,
the generalized resolvent of the relation L0.
20. Let M (λ)be the characteristic operator of the equation (1) (and therefore by [20]
ℑM (λ) ≥ 0 as ℑλ > 0) and σ (µ) = w− lim
ε↓0
1
π
∫ µ
0 ℑM (µ + iε) dµ be the spectral operator-
function that corresponds to M (λ).
Let Eµ be the generalized spectral family corresponding to the generalized resolvent Rλ
from the item 1◦ of this theorem. Then for any f(t) ∈ Cs
0
(
Ī
)
the equality
(28)
◦
P Eα, βf(t) =
◦
P
∫ β
α
[Xµ(t)]1 dσ (µ) ϕ (µ, f) ,
is valid in
◦
L2
m (I), where [Xλ(t)]1 ∈ B (Hp, H) is the first row of the operator solution
Xλ(t) of the homogeneous equation (2) with coefficients (10), (11) that is written in the
matrix form and such that Xλ (0) = Ip,
(29) ϕ (µ, f) =
∫
I
(
[Xµ(t)]1
)∗
m [f ]dt,
if p−1 (t, µ) ∈ B (H) ∀t ∈ Ī, µ ∈ [α, β].
Moreover, for f(t) ∈ D(L′
0) (see (26)) and with r > s (or with r < s, if additionally
◦
P (E+0 − E0) f(t) = 0), the inverse formula in
◦
L2
m (I)
(30) f(t) =
◦
P
∫ ∞
−∞
[Xµ(t)]1 dσ (µ)ϕ (µ, f) ,
and Parceval’s equality
(31) m [f(t), g(t)] = (ϕ (µ, f) , ϕ (µ, g))L2(R, dσ) ,
are valid, where g(t) ∈ Cs
0
(
Ī
)
.
Let us explain that, for r > s,
◦
P
∫ ∞
−∞
= lim
α→−∞
β→∞
◦
P
∫ β
α
EXPANSION IN EIGENFUNCTIONS OF RELATIONS 145
in (30), and for r < s,
◦
P
∫ ∞
−∞
= lim
α→−∞
β→−0
◦
P
∫ β
α
+ lim
δ→∞
γ→+0
◦
P
∫ δ
γ
,
where the limits exist in
◦
L2
m (I). Similarly,
∫∞
−∞
=
∫ −0
−∞
+
∫∞
+0 in the right-hand side of
(31) for r < s.
Proof. Let for definiteness r ≤ s = 2n (for r > s the proof becomes simpler due to
(10)–(12)).
10. Let ℑλ �= 0. In view of the item 1◦ of Lemma 3, y1 (t, λ, f) is a solution of (1).
Using (10) and Lemmas 1–3 one can show that
(32)
s∑
k=0
mk [y1 (t, λ, f) , y1 (t, λ, f)] −ℑ
( s∑
k=0
mk [y1 (t, λ, f) , f(t)]
)
/ℑλ
=
(
Wλ(t)x (t, λ, f) , x(t, λ, f)
)
Hs −ℑ
(
Wλ(t)x (t, λ, f) , Fλ̄(t)
)
Hs/ℑλ,
although for r ≤ s the corresponding items in the right- and left-hand sides of (32) do
not coincide. Therefore1
(33)
‖y1 (t, λ, f)‖2
L2
m(α, β) −ℑ (y1 (t, λ, f) , f(t))L2
m(α, β) /ℑλ
= ‖x (t, λ, f)‖2
L2
Wλ
(α, β) −ℑ (x (t, λ, f) , Fλ̄(t))L2
Wλ
(α, β) /ℑλ.
In view of the item 2◦ of Lemma 3 a nonnegative limit of the right-hand-side of (33)
exists, when (α, β) ↑ I. Consequently
(34) ‖y1 (t, λ)‖2
L2
m(I) ≤ ℑ (y1 (t, λ) , f(t))L2
m(I) /ℑλ.
Since M (λ) = M∗
(
λ̄
)
, then the operator Rλ (3) in L2
Wλ
(α, β) with finite (α, β) ⊆ I
possesses the property Rλ = R∗
λ̄
. Therefore ([ℜQ(t)] RλF, Rλ̄G)|βα = 0. It follows from
Corollary 1 and (34) that ∀f(t), g(t) ∈ Cs
(
Ī
)⋂
L2
m (I)
m [y1 (λ, f) , g] = m
[
f, y1
(
λ̄, g
)]
.
Thus the closure of the operator Rλf = y1 (t, λ, f) in L2
m (I) possesses a property
(35) Rλ = R∗
λ̄.
Since in view of (34) for any f(t), g(t) ∈ Cs
(
Ī
)⋂
L2
m (I) and with (α, β) ↑ I,
(y1 (λ, f) , g)L2
m(α, β) → (y1 (λ, f) , g)L2
m(I)
uniformly in λ from any compact set ∈ C/R, we see that, in view of analyticity of the
operator function M (λ) and vector-function Wλ(t)Fλ̄(t), (17), the operator Rλ depends
analytically on the non-real λ in view of [15, p. 195].
Finally, similarly to the case s = 0 [20] using Corollary 1 it is verified that
Rλ (L0 − λ) ⊂ I,(36)
where I is the graph of the identical operator in L2
m (I).
Taking into account (34)–(36) and analyticity of Rλ, we see in view of [10] that Rλ is
a generalized resolvent of L0. Item 1◦ is proved.
1In particular this implies that
‖y1 (t, λ, f)‖2
L2
m(α,β) −
ℑ (y1 (t, λ, f) , f(t))L2
m(α,β)
ℑλ
=
U [x (β, λ, f)] − U [x (α, λ, f)]
2ℑλ
.
146 VOLODYMYR KHRABUSTOVSKYI
20. Let the vector-functions f(t), g(t) ∈ Cs
0 (I) , λ = µ + iε, Gλ(t) be defined by (16)
with g(t) instead of f(t). In view of the Stieltjes inversion formula,
(37)
(Eα, β f, g)m = lim
ε↓0
1
2πi
∫ β
α
([
y1 (λ, f) − y1
(
λ̄, f
)]
, g
)
m
dµ
= lim
ε↓0
1
2πi
∫ β
α
[
(x (t, λ, f) , Gλ(t))L2
Wλ
(I) −
(
x
(
t, λ̄, f
)
, Gλ̄(t)
)
L2
Wλ
(I)
+2i
∫
I
((
ℑp−1 (t, λ)
)
f [n] (t|m) , g[n] (t|m)
)
dt
]
dµ
= lim
ε↓0
1
2πi
∫ β
α
[(
M (λ)
∫
I
X∗
λ̄(t)Wλ(t)Fλ̄(t)dt,
∫
I
X∗
λ(t)Wλ̄(t)Gλ(t)dt
)
−
(
M∗ (λ)
∫
I
X∗
λ(t)Wλ̄(t)Fλ(t)dt,
∫
I
X∗
λ̄(t)Wλ(t)Gλ̄(t)dt
)]
dµ
=
∫ β
α
(
dσ (µ)
∫
I
X∗
µ(t)Wµ(t)Fµ(t)dt,
∫
I
X∗
µ(t)Wµ(t)Gµ(t)dt
)
,
where the second equality is a corollary of (10), (13), (20), (23), next to last is a corollary
of (3), and the last follows from the well-known generalization of the Stieltjes inversion
formula [27, proposition (�), p. 803], [4, Lemma, p. 952]. But for µ ∈ [α, β]
(38)
∫
I
X∗
µ(t)Wµ(t)Fµ(t)dt =
∫
I
(
[Xµ(t)]1
)∗
m [f ]dt,
because, in view of (20),
∀h ∈ Hs :
(∫
I
X∗
µ(t)Wµ(t)Fµ(t)dt, h
)
=
∫
I
(Wµ(t)Fµ(t), Xµ(t)h) =
∫
I
( (
[Xµ]1
)∗
m [f ] , h
)
dt.
Due to (37), (38),
(39) (Eα, βf, g)m =
∫ β
α
(dσ (µ)ϕ (µ, f) , ϕ (µ, g)) .
Replacing
∫ β
α in (39) by an integral sum and using (20), (38) we obtain that
(Eα, βf, g)m =
(∫ β
α
[Xµ(t)]1 dσ (µ)ϕ (µ, f) , g(t)
)
m
=
(
◦
P
∫ β
α
[Xµ(t)]1 dσ (µ)ϕ (µ, f) , g(t)
)
m
and (28) is proved.
Since E∞f(t) = f(t) if f(t) ∈ D (L′
0), passing to the limit in (28), (39) for α →
−∞, β → −0 and α → +0, β → ∞ we obtain (30) and (31). Item 2◦ and Theorem 1
are proved. �
The following remark follows from [5, 6] and from [20, formula (1.70)].
Remark 1. If m[y] = w(t)y and if for the equation (2) with coefficients (10), (11) that
corresponds to the equation (6), the condition
∃λ0 ∈ C, α, β, δ > 0 : (∆λ0
(α, β) g, g) ≥ δ ‖g‖2 ∀g ∈ N⊥(40)
holds true, then Rλf for any generalized resolvent Rλ of L0 and any f(t) ∈ C0(Ī) have
the same representation as in item 1◦ of Theorem 1.
An analysis of the proof of Theorem 1 shows the following.
EXPANSION IN EIGENFUNCTIONS OF RELATIONS 147
Remark 2. If (14) is not assumed to hold, then we have the following: 1) item 1◦ of
Theorem 1 is valid either for f(t) ∈ Cs
(
Ī
)
, if the interval I is finite or for f(t) ∈ Cs
0 (I),
if
◦
L2
m (I) = L2
m (I).
2) Identity (28) holds for L2
m (I), if one changes it as follows: a) σ (µ) is a spectral
function corresponding to PM (λ) P (ℑPM (λ)P ≥ 0 as ℑλ > 0 [20]); b) remove
◦
P from
(28); c) f(t) ∈ Cs
(
Ī
)
and ϕ (µ, f) =
∫
I
X∗
µ(t)Wµ(t)Fµ(t)dt, if the interval I is finite or
f(t) ∈ Cs
0
(
Ī
)
, if
◦
L2
m (I) = L2
m (I).
The following theorem establishes a relationship between the generalized resolvents of
the relations L0 that are given by Theorem 1, and the boundary value problems for the
equation (1) with boundary conditions depending on the spectral parameter. Already in
the simplest case, where l and m that generate (1) are self-adjoint differential operators we
see that the pair {y, f} satisfies the boundary conditions that contain both y derivatives
and f derivatives of corresponding orders at the ends of interval.
Theorem 2. Let the interval I = (a, b) be finite.
Let the operator-functions Mλ, Nλ ∈ B (Hp), depend analytically on the non-real λ,
(41) M∗
λ̄ [ℜQ (a)]Mλ = N ∗
λ̄ [ℜQ (b)]Nλ (ℑλ �= 0),
where Q(t) is the coefficient of the equation (2) corresponding by Lemma 3 to the equation
(1) (see (10 ), (11)),
(42) ‖Mλh‖ + ‖Nλh‖ > 0 (0 �= h ∈ Hp, ℑλ �= 0) ,
the lineal {Mλh ⊕Nλh |h ∈ Hp } ⊂ H2p is a maximal Q-nonnegative subspace since
ℑλ �= 0, where Q = (ℑλ) diag (ℜQ (a) ,−ℜQ (b)) (and therefore
(43) ℑλ (N ∗
λ [ℜQ (b)]Nλ −M∗
λ [ℜQ (a)]Mλ) ≤ 0 (ℑλ �= 0)).
Then for any f(t) ∈ Cs
(
Ī
)
the boundary problem that is obtained by adding the
boundary conditions
(44) ∃h = h (λ, f) ∈ Hp : x (a, λ, f) = Mλh, x (b, λ, f) = Nλh,
to the equation (1), where x (t, λ, f) is defined by (23), has the unique solution Rλf
as ℑλ �= 0. It is generated by the generalized resolvent Rλ of the relation L0 that is
constructed, as in item 1◦ of Theorem 1, using the c.o.
M (λ) = −1
2
(
X−1
λ (a)Mλ + X−1
λ (b)Nλ
) (
X−1
λ (a)Mλ − X−1
λ (b)Nλ
)−1
(iG)−1 ,
where (
X−1
λ (a)Mλ − X−1
λ (b)Nλ
)−1 ∈ B (Hp) (ℑλ �= 0) ,
Xλ(t) is an operator solution of the homogeneous equation (2) with coefficients (10), (11)
and such that Xλ (0) = Ip.
Proof. Proof follows from Lemma 3, Theorem 1 and from from [20, Remark 1.1]. �
For s = 0, Theorem 2 is known (see [28, 5] as dimH < ∞, [20] as r = 1, dimH = ∞).
Example 2. Let, in the equation (1), r = 4, s = 2.
a) Let Mλ = Nλ = Ip. Then the boundary conditions (44) can be represented in the
form
y (a) = y (b) , y′ (a) = y′ (b) ,
y[2] (a |l ) = y[2] (b |l ) ,
y[3] (a |l − λm ) − f [1] (a |m ) = y[3] (b |l − λm ) − f [1] (b |m ) .
(45)
In particular for the equation
y(IV ) − λ (−y′′ + y) = −f ′′ + f(46)
148 VOLODYMYR KHRABUSTOVSKYI
conditions (45) have the form
y (a) = y (b) , y′ (a) = y′ (b) , y′′ (a) = y′′ (b) ,
y′′′ (a) + f ′ (a) = y′′′ (b) + f ′ (b) .
(47)
b) If dimH = 1,
Mλ =
0 0 0 0
0 1 0 0
0 0 1 0
0 0 0 1
, Nλ =
0 0 0 0
0 0 0 −1
1 0 0 0
0 1 0 0
,
then the boundary conditions (44) can be written in the form
y(a) = y(b) = 0, y′(a) = y[2](b|l), y[2](a|l) = −y′(b).(48)
In particular for the equation (46) conditions (48) have the form
y(a) = y(b) = 0, y′(a) = y′′(b), y′′(a) = −y′(b).(49)
On functions satisfying either the boundary condition (47) with f(t) ≡ 0 or the bound-
ary conditions (49), expressions l[y] = y(IV) and m[y] = −y′′ + y define a self-adjoint and
symmetric operators correspondingly.
When the boundary conditions are such that l[y] defines a self-adjoint operator and
m[y] defines only a symmetric operator on functions satisfying these conditions, then
the eigenfunction expansion of the special scalar equation (6) in the regular case is
constructed in [7, 14]. We note that, in the general case, the boundary conditions (44) are
not reduced to the boundary condition of [7, 14] type, since it is possible that conditions
(44) in the case [7, 14] do not imply s boundary conditions containing only the derivatives
of order up to s − 1.
In the next theorem, I = R and condition (14) hold both on the negative semi-axis
R− (i.e. as I = R+) and on the positive semi-axis R+ (i.e. as I = R−).
Theorem 3. Let I = R, the coefficient of the equation (6) be periodic on each of the
semi-axes R+ and R− with periods T+ > 0 and T− > 0 correspondingly. Then the
spectrums of the monodromy operators Xλ (±T±) (Xλ(t) is from Theorem 2 ) do not
intersect the unit circle as ℑλ �= 0, the c.o. M (λ) of the equation (1) is unique and equal
to
(50) M (λ) =
(
P (λ) − 1
2
Ip
)
(iG)
−1
(ℑλ �= 0)
where the projection P (λ) = P+ (λ) (P+ (λ) + P− (λ))
−1
, P± (λ) are Riesz projections of
the monodromy operators Xλ (±T±) that correspond to their spectrums lying inside the
unit circle, (P+ (λ) + P− (λ))
−1 ∈ B (Hp) as ℑλ �= 0.
Also let dimH < ∞, A = {µ ∈ R : det p (t, µ) �= 0 ∀t ∈ (−T−, T+)}, a finite interval
∆ ⊂ A. Then in item 2◦ of Theorem 1 dσ (µ) = dσac (µ) + dσd (µ) , µ ∈ ∆. Here
σac (µ) ∈ AC(∆) and, for µ ∈ ∆,
(51) σ′
ac (µ) =
1
2π
G−1
(
Q∗
− (µ)GQ− (µ) − Q∗
+ (µ)GQ+ (µ)
)
G−1
where the projections Q± (µ) = q± (µ) (P+ (µ) + P− (µ))
−1
, q± (µ) are Riesz projections
of the monodromy matrixes Xµ (±T±) corresponding to the multiplicators equal to 1 such
that they are shifted inside the unit circle as µ is shifted to the upper half plane, P± (µ) =
P± (µ + i0)); σd (µ) is a jump function.
Let us notice that the sets on which q± (µ) , P± (µ) , (P+ (µ) + P− (µ))
−1
are not
infinitely differentiable do not have finite limit points ∈ A as well as the set of points of
increase of σd (µ).
EXPANSION IN EIGENFUNCTIONS OF RELATIONS 149
Proof. Let the operator G be indefinite (otherwise the proof is modified in an obvious
way). The unitary dichotomy of the operators Xλ (±T±) and the fact that M (λ) (50) is a
c.o. of the equation (1) on (−T−, T+) follow from [20, p. 161, 162]. Since Xλ (t ± T±) =
Xλ(t)Xλ (±T±), t ∈ R±, and ℑλU [Xλ(t)] does not decrease as ℑλ �= 0, we have that
M (λ) (50) is a c.o. of the equation (1) on any finite I and therefore it is a c.o. on the
axis.
Let for some non-real λ0 the homogeneous equation (2) with coefficients (10), (11)
have a solution x(t) ∈ L2
Wλ0
(
R1
)
.
Since, for k ∈ Z+,
‖x(t)‖2
L2
Wλ0
(R1) =
0∑
j=−∞
(∆λ0
(−kT−, 0)x (−jkT−) , x (−jkT−))
+
∞∑
j=0
(∆λ0
(0, kT+)x (jkT+) , x (jkT+)) ,
and using condition (14) on I = R± and estimates of the type [8, p. 290], we see that
x (0) ∈ H−
⋂
H+, where H± are the invariant subspaces of the operators Xλ (±T±) that
correspond to their spectrums lying inside the unit circle. But H−
⋂
H+ = {0} [20,
p. 162]. Therefore in view of Lemma 1.5 from [20] the c.o. M (λ) (50) is unique.
Formula (51) follows from [19, Theorem 13]. Decomposition dσ (µ) = dσac (µ) + dσd,
µ ∈ ∆ as well as the remark after the formulating of Theorem 3 are proved in the same
way as similar statements in [18]. In the proof for r ≤ s one should take into account
that Krein-Lyubarsky theory [22] for homogeneous periodic system (2) is still valid for
λ ∈ A⋂
R and when Hλ(t) contains λ is a Nevanlinna manner; it can be seen analysing
the statement of this theory in [29, p. 147–150, 181–183] and the proof of Theorem 1.2
from [12, p. 305]. Theorem is proved. �
Example 3. Let dimH = 1, l[y] = (i)
n
y(n), m[y] = (i)
2n
y(2n) + y, I = R (and
therefore L2
m (I) =
◦
L2
m (I)). In this case, E0 = E+0, the spectral matrix σ (µ) ∈ ACloc,
and, in view of Theorem 3 for n = 1, 3, . . .,
(52)
σ′ (µ) =
1
2πn (k2n + 1)
{
2knAn + (1 − k2n)I2n +
n−1∑
j=1
(
k2n−j
+ (−1)
j+1
kj
) (
Aj + (−1)j+1A−j
)}
(iJ)
−1
,
since |µ| < 1
2 , where
∑0
j=1 = 0, rgσ′ (µ) = 2, and σ′ (µ) = 0 as |µ| > 1
2 . For n =
2, 4, 6, . . ., one has
(53) σ′ (µ) =
1
πn (k2n − 1)
n/2∑
j=1
(
k2n−2j+1 − k2j−1
) (
A2j−1 + A1−2j
)
(iJ)
−1
since 0 < µ < 1
2 , where rgσ′ (µ) = 4, and σ′ (µ) = 0 as µ /∈ [0, 1/2]. In (52), (53) k =
−i
n
√
1+(−1)n
√
1−4µ2
2µ , A = (iJ)
−1
Hµ(t), where the matrices J and Hµ(t) are independent
of t and defined by (10).
In particular, for n = 1, |µ| < 1/2 ,
σ′ (µ) =
1
2π
(
2√
1−4µ2
0
0 1
2
√
1 − 4µ2
)
.
150 VOLODYMYR KHRABUSTOVSKYI
And for n = 2, 0 < µ < 1
2 ,
σ′ (µ) =
1
π
√
1 +
√
1 − 4µ2
2µ (1 − 2µ)
· 1√
1 − 2µ +
√
1 + 2µ
1 0 0 µ
0 1 µ 0
0 µ µ (1 − µ) 0
µ 0 0 µ (1 − µ)
.
Remark 3. In the case r ≤ s in contrast to the case r > s, the point spectrum of
the relation L′
0 can be non-empty including the case when L′
0 corresponds to the scalar
equation (1) with periodic coefficients on the axis.
Indeed let m[y] = −y′′ + y, l[y] = p(t)y, where p (t + 4) = p(t) ∈ C (R), p(t) ={
1, 4k ≤ t ≤ 4k + 1
0, 4k + 2 ≤ t ≤ 4k + 3
. Then for any function y(t) ∈ C2
0 (R) such that suppy(t) ⊂
⋃
k
[4k + 2, 4k + 3], the pair
{
y(t), 0
}
∈ L′
0, i.e. the point spectrum of L′
0 contains λ = 0.
Similarly an example for r = 1, r = 2, s = 2 is constructed.
The following remark is proved similarly to Theorem 3.
Remark 4. Let I = R+, the coefficients of the equation (6) be periodic with the period
T > 0. Then
1) Any c.o. of the equation (1) is found by combining the formula (4.4) from [20] and
the formula (50).
2) If ∞ > dimHp = 2k, L is a k dimensional G-neutral subspace (see [8]), then
the c.o. of the equation (1), which corresponds to a self-adjoint boundary condition in
zero, x (0, λ, f) ∈ L, is given by the formula (36) from [18]. The corresponding spectral
matrix-function σ(µ)(dσac(µ), µ ∈ ∆+) is given for r > s (r ≤ s) by Theorem 6 from
[18], starting from the monodromy matrix Xλ (T ) of the equation (6), where ∆+ is an
analog of ∆ from Theorem 3.
Acknowledgments. The author is grateful to Professor F. S. Rofe-Beketov for his great
attention to this work.
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Ukrainian State Academy of Railway Transport, 7 Feyerbakh square, Kharkiv, 61050,
Ukraine
E-mail address: v khrabustovskyi@ukr.net
Received 19/03/2009
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