Hindawi Publishing Corporation
Abstract and Applied Analysis
Volume 2016, Article ID 2964817, 6 pages
http://dx.doi.org/10.1155/2016/2964817
Research Article
Resolvent for Non-Self-Adjoint Differential Operator with
Block-Triangular Operator Potential
Aleksandr Mikhailovich Kholkin
Department of Higher and Applied Mathematics, Priazovskyi State Technical University, Universitetskaya Street 7,
Mariupol 87500, Ukraine
Correspondence should be addressed to Aleksandr Mikhailovich Kholkin; [email protected]
Received 10 July 2016; Revised 30 September 2016; Accepted 6 October 2016
Academic Editor: Cemil Tunç
Copyright © 2016 Aleksandr Mikhailovich Kholkin. This is an open access article distributed under the Creative Commons
Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is
properly cited.
A resolvent for a non-self-adjoint differential operator with a block-triangular operator potential, increasing at infinity, is
constructed. Sufficient conditions under which the spectrum is real and discrete are obtained.
1. Introduction
2. Preliminary Notes
The theory of non-self-adjoint singular differential operators,
generated by scalar differential expressions, has been well
studied. An overview on the theory of non-self-adjoint
singular ordinary differential operators is provided in V. E.
Lyantse’s Appendix I to the monograph of Naimark [1]. In
this regard the papers of Naimark [2], Lyantse [3], Marchenko
[4], Rofe-Beketov [5], Schwartz [6], and Kato [7] should
be noted. The questions regarding equations with nonHermitian matrix or operator coefficients have been studied
insufficiently. For a differential operator with a triangular
matrix potential decreasing at infinity, which has a bounded
first moment due to the inverse scattering problem, it is stated
in [8, 9] that the discrete spectrum of the operator consists
of a finite number of negative eigenvalues, and the essential
spectrum covers the positive semiaxis. The questions regarding an operator with a block-triangular matrix potential that
increases at infinity are considered in [10, 11]. In the future,
by the author of this paper similar questions are considered
for equations with block-triangular operator coefficients. In
[11, 12] Green’s function of a non-self-adjoint operator is
constructed.
In this article we construct a resolvent for a non-selfadjoint differential operator, using which the structure of the
operator spectrum is set.
Let 𝐻𝑘 , 𝑘 = 1, 2, . . . , 𝑟, be finite-dimensional or infinitedimensional separable Hilbert space with inner product (⋅, ⋅)
and norm | ⋅ |, dim 𝐻𝑘 ≤ ∞. Denote H = 𝐻1 ⊕ 𝐻2 ⊕
⋅ ⋅ ⋅ ⊕ 𝐻𝑟 . Element ℎ ∈ H will be written in the form ℎ =
col (ℎ1 , ℎ2 , . . . , ℎ𝑟 ), where ℎ𝑘 ∈ 𝐻𝑘 , 𝑘 = 1, 𝑟, 𝐼𝑘 , 𝐼 are identity
operators in 𝐻𝑘 and H accordingly.
We denote by 𝐿 2 (H, (0, ∞)) the Hilbert space of vectorvalued functions 𝑦(𝑥) with values in H with inner product
∞
⟨𝑦, 𝑧⟩ = ∫ (𝑦 (𝑥) , 𝑧 (𝑥)) 𝑑𝑥
(1)
0
and the corresponding norm ‖ ⋅ ‖.
Consider the equation with block-triangular operator
potential
𝑙 [𝑦] = −𝑦󸀠󸀠 + 𝑉 (𝑥) 𝑦 = 𝜆𝑦, 0 ≤ 𝑥 < ∞,
(2)
where
𝑉 (𝑥) = V (𝑥) ⋅ 𝐼 + 𝑈 (𝑥) ,
𝑈11 (𝑥) 𝑈12 (𝑥) ⋅ ⋅ ⋅ 𝑈1𝑟 (𝑥)
𝑈 (𝑥) = (
0
𝑈22 (𝑥) ⋅ ⋅ ⋅ 𝑈2𝑟 (𝑥)
⋅⋅⋅
⋅⋅⋅
⋅⋅⋅
⋅⋅⋅
0
0
⋅ ⋅ ⋅ 𝑈𝑟𝑟 (𝑥)
),
(3)
2
Abstract and Applied Analysis
V(𝑥) is a real scalar function, and 0 < V(𝑥) → ∞ monotonically, as 𝑥 → ∞, and it has monotone absolutely continuous
derivative. Also, 𝑈(𝑥) is a relatively small perturbation; for
example, |𝑈(𝑥)| ⋅ V−1 (𝑥) → 0 as 𝑥 → ∞ or |𝑈|V−1 ∈ 𝐿∞ (R+ ).
The diagonal blocks 𝑈𝑘𝑘 (𝑥), 𝑘 = 1, 𝑟, are assumed as bounded
self-adjoint operators in 𝐻𝑘 , 𝑈𝑘𝑙 : 𝐻𝑙 → 𝐻𝑘 .
In case where
V (𝑥) ≥ 𝐶𝑥2𝛼 , 𝐶 > 0, 𝛼 > 1,
(4)
we suppose that coefficients of (2) satisfy relations
∞
∞
(5)
∞
𝑛
𝑥
1
⋅ exp (∫ √V (𝑢)𝑑𝑢) .
4
√4V (𝑥)
0
−𝑧󸀠󸀠 + (V (𝑥) + 𝑞 (𝑥)) 𝑧 = 0,
(7)
where 𝑞(𝑥) is determined by a formula (cf. with the monograph [13])
2
1 V󸀠󸀠 (𝑥)
5 V󸀠 (𝑥)
(
) −
.
16 V (𝑥)
4 V (𝑥)
(8)
(9)
In case of V(𝑥) = 𝑥2𝛼 , 0 < 𝛼 ≤ 1, we suppose that the
coefficients of (2) satisfy the relation
−𝛼
∫ |𝑈 (𝑡)| ⋅ 𝑡 𝑑𝑡 < ∞, 𝑎 > 0.
𝑎
(10)
Now functions 𝛾0 (𝑥, 𝜆) and 𝛾∞ (𝑥, 𝜆) are defined as follows:
𝛾0 (𝑥, 𝜆) =
𝛾∞ (𝑥, 𝜆) =
√4 (𝑥2𝛼
4
1
√4 (𝑥2𝛼 − 𝜆)
4
⋅ exp (− ∫ √𝑢2𝛼 − 𝜆𝑑𝑢) ,
𝑎
𝑥
⋅ (1 + 𝑜 (1)) .
𝛾0 (𝑥, 𝜆) = 𝑐 ⋅ 𝑥(𝜆−1)/2 ⋅ exp (−
𝛾∞ (𝑥, 𝜆) = 𝑐 ⋅ 𝑥
−(𝜆+1)/2
𝑥2
) (1 + 𝑜 (1)) ,
2
𝑥2
⋅ exp ( ) (1 + 𝑜 (1)) .
2
(13)
In the case (𝛼 + 1)/2𝛼 ∉ N, set 𝑛 = [(𝛼 + 1)/2𝛼] + 1, with [𝛽]
being the integral part of 𝛽, to obtain the following asymptotic
behavior for 𝛾0 (𝑥, 𝜆) and 𝛾∞ (𝑥) at infinity:
𝛾0 (𝑥, 𝜆) = 𝑐 ⋅ 𝑥−𝛼/2 exp (−
𝑥1+𝛼 𝜆 𝑥1−𝛼
+ ⋅
1+𝛼 2 1−𝛼
1 ⋅ 3 ⋅ . . . ⋅ (2𝑘 − 3)
𝑥1−(2𝑘−1)𝛼
𝜆 𝑘
⋅( ) ⋅
)
𝑘!
2
1 − (2𝑘 − 1) 𝛼
𝑘=2
𝑛−1
⋅ exp (−
𝜆 𝑛 𝑥−𝛼
1 ⋅ 3 ⋅ . . . ⋅ (2𝑛 − 3)
⋅( ) ⋅
) ⋅ (1
𝑛!
2
𝛼
+ 𝑜 (𝑥−𝛼 )) ,
𝑥1+𝛼 𝜆 𝑥1−𝛼
𝛾∞ (𝑥, 𝜆) = 𝑐 ⋅ 𝑥−𝛼/2 exp (
− ⋅
1+𝛼 2 1−𝛼
(14)
1 ⋅ 3 ⋅ . . . ⋅ (2𝑘 − 3)
𝑥1−(2𝑘−1)𝛼
𝜆 𝑘
⋅( ) ⋅
)
𝑘!
2
1 − (2𝑘 − 1) 𝛼
𝑘=2
𝑛−1
−∑
𝑥
− 𝜆)
+𝛼/2)
+∑
In such a way for all 𝑥 ∈ [0, ∞) one has
󸀠
𝑊 (𝛾0 , 𝛾∞ ) fl 𝛾0 (𝑥) ⋅ 𝛾∞
(𝑥) − 𝛾0󸀠 (𝑥) ⋅ 𝛾∞ (𝑥) = 1.
(12)
In particular, with 𝛼 = 1 (𝑛 = 1) one has
(6)
It is easy to see that 𝛾0 (𝑥) → 0, 𝛾∞ (𝑥) → ∞ as 𝑥 → ∞. These
solutions constitute a fundamental system of solutions of the
scalar differential equation
1
𝑥1+𝛼 𝜆 𝑥1−𝛼
𝛾∞ (𝑥, 𝜆) = 𝑐 ⋅ exp (
− ⋅
1+𝛼 2 1−𝛼
⋅ 𝑥−(((1⋅3⋅...⋅(2𝑛−3))/𝑛!)⋅(𝜆/2)
𝑥
1
⋅ exp (− ∫ √V (𝑢)𝑑𝑢) ,
𝛾0 (𝑥) = 4
√4V (𝑥)
0
⋅ (1 + 𝑜 (1)) ,
1 ⋅ 3 ⋅ . . . ⋅ (2𝑘 − 3)
𝑥1−(2𝑘−1)𝛼
𝜆 𝑘
⋅( ) ⋅
)
𝑘!
2
1 − (2𝑘 − 1) 𝛼
𝑘=2
Let us consider the functions
∞
−𝛼/2
−∑
0
𝑞 (𝑥) =
1 ⋅ 3 ⋅ . . . ⋅ (2𝑘 − 3)
𝑥1−(2𝑘−1)𝛼
𝜆 𝑘
⋅( ) ⋅
)
𝑘!
2
1 − (2𝑘 − 1) 𝛼
𝑘=2
𝑛−1
+∑
𝑛−1
∫ V󸀠󸀠 (𝑡) ⋅ V−3/2 (𝑡) 𝑑𝑡 < ∞.
𝛾∞ (𝑥) =
𝑥1+𝛼 𝜆 𝑥1−𝛼
+ ⋅
1+𝛼 2 1−𝛼
𝑛
0
0
𝛾0 (𝑥, 𝜆) = 𝑐 ⋅ exp (−
⋅ 𝑥((1⋅3⋅...⋅(2𝑛−3))/𝑛!)⋅(𝜆/2)
∫ |𝑈 (𝑡)| ⋅ V−1/2 (𝑡) 𝑑𝑡 < ∞,
∫ V󸀠2 (𝑡) ⋅ V−5/2 (𝑡) 𝑑𝑡 < ∞,
In [10] the asymptotic behavior of the functions 𝛾0 (𝑥, 𝜆)
and 𝛾∞ (𝑥, 𝜆) was established as 𝑥 → ∞. If (𝛼+1)/2𝛼 = 𝑛 ∈ N,
that is, 𝛼 = 1/(2𝑛 − 1), then functions 𝛾0 (𝑥, 𝜆) and 𝛾∞ (𝑥, 𝜆)
as 𝑥 → ∞ will have the following asymptotic behavior:
(11)
⋅ exp (∫ √𝑢2𝛼 − 𝜆𝑑𝑢) .
𝑎
These functions also form a fundamental system of solutions
of the scalar differential equation, which is obtained by
replacing V(𝑥) with V(𝑥) − 𝜆 in formulas (7) and (8).
⋅ exp (
𝜆 𝑛 𝑥−𝛼
1 ⋅ 3 ⋅ . . . ⋅ (2𝑛 − 3)
⋅( ) ⋅
) ⋅ (1
𝑛!
2
𝛼
+ 𝑜 (𝑥−𝛼 )) .
In [10] for an equation with matrix coefficients, and in
the furtherance for equations with operator coefficients, the
following theorem is proved.
Abstract and Applied Analysis
3
Theorem 1. If for (2) conditions (4)-(5) are satisfied for 𝛼 > 1
or condition (10) for 0 < 𝛼 ≤ 1, then the equation has a unique
decreasing at infinity operator solution Φ(𝑥, 𝜆), satisfying the
conditions
lim
Φ (𝑥, 𝜆)
= 𝐼,
(𝑥, 𝜆)
𝑥→∞ 𝛾
0
(15)
Φ󸀠 (𝑥, 𝜆)
lim 󸀠
= 𝐼.
𝑥→∞ 𝛾 (𝑥, 𝜆)
0
Also, there exists increasing at infinity operator solution
Ψ(𝑥, 𝜆), satisfying the conditions
lim
Ψ (𝑥, 𝜆)
= 𝐼,
(𝑥, 𝜆)
(16)
Ψ󸀠 (𝑥, 𝜆)
lim 󸀠
= 𝐼.
𝑥→∞ 𝛾 (𝑥, 𝜆)
∞
𝛾∞ (𝑥, 𝜆) = 𝑥
2
𝑥
),
2
𝑥2
⋅ exp ( ) .
2
(17)
3. Resolvent of the Non-Self-Adjoint Operator
Let the following boundary condition be given at 𝑥 = 0:
󸀠
cos 𝐴 ⋅ 𝑦 (0) − sin 𝐴 ⋅ 𝑦 (0) = 0,
(18)
where 𝐴 is block-triangular operator of the same structure
as the potential 𝑉(𝑥) (3) of the differential equation (2), and
𝐴 𝑘𝑘 , 𝑘 = 1, 𝑟, are the bounded self-adjoint operators in 𝐻𝑘 ,
which satisfy the conditions
(19)
Together with problem (2) and (18) we consider the separated
system
𝑙𝑘 [𝑦𝑘 ] = −𝑦𝑘󸀠󸀠 + (V (𝑥) 𝐼𝑘 + 𝑈𝑘𝑘 (𝑥)) 𝑦𝑘 = 𝜆𝑦𝑘 ,
𝑘 = 1, 𝑟
(20)
(0) − sin 𝐴 𝑘𝑘 ⋅ 𝑦𝑘 (0) = 0,
Comment 4. Note that this theorem contains a statement of
the discrete spectrum of the non-self-adjoint operator 𝐿 only
and no allegations of its continuous and residual spectrum.
Along with (2) we consider the equation
𝑙1 [𝑦] = −𝑦󸀠󸀠 + 𝑉∗ (𝑥) 𝑦 = 𝜆𝑦
(23)
(𝑉∗ (𝑥) is adjoint to the operator 𝑉(𝑥)). If the space H is finitedimensional, then (23) can be rewritten as
̃𝑙 [̃
(24)
̃ 𝑉 (𝑥) = 𝜆̃
𝑦] = −̃
𝑦󸀠󸀠 + 𝑦
𝑦,
̃ 𝑟 ) and the equation is called the left.
̃2 . . . 𝑦
𝑦1 𝑦
̃ = (̃
where 𝑦
For operator functions 𝑌(𝑥, 𝜆), 𝑍(𝑥, 𝜆) ∈ 𝐵(H) let
𝑊 {𝑍∗ , 𝑌} = 𝑍∗󸀠 (𝑥, 𝜆) 𝑌 (𝑥, 𝜆)
− 𝑍∗ (𝑥, 𝜆) 𝑌󸀠 (𝑥, 𝜆) .
(25)
If 𝑌(𝑥, 𝜆) is operator solution of (2) and 𝑍(𝑥, 𝜆) is operator
solution of (23), the Wronskian does not depend on 𝑥.
Now we denote 𝑌(𝑥, 𝜆) and 𝑌1 (𝑥, 𝜆) as the solutions of (2)
and (23), respectively, satisfying the initial conditions
𝑘 = 1, 𝑟.
𝑌󸀠 (0, 𝜆) = sin 𝐴,
𝑌1 (0, 𝜆) = (cos 𝐴)∗ ,
(26)
𝑌1󸀠 (0, 𝜆) = (sin 𝐴)∗ ,
𝜆 ∈ C.
Because the operator function 𝑌1∗ (𝑥, 𝜆) satisfies equation
with the boundary conditions
cos 𝐴 𝑘𝑘 ⋅
(22)
𝑌 (0, 𝜆) = cos 𝐴,
𝜋
𝜋
− 𝐼𝑘 ≪ 𝐴 𝑘𝑘 ≤ 𝐼𝑘 .
2
2
𝑦𝑘󸀠
𝜎𝑑 (𝐿) = ⋃ 𝜎 (𝐿 𝑘 ) .
𝑘=1
Corollary 2. If 𝛼 = 1, that is, V(𝑥) = 𝑥2 , then, under condition
(10), the solutions Φ(𝑥, 𝜆) and Ψ(𝑥, 𝜆) have common (known)
asymptotic behavior, as in the quality 𝛾0 (𝑥, 𝜆) and 𝛾∞ (𝑥, 𝜆) you
can take the following functions:
−(𝜆+1)/2
Theorem 3. Suppose that for (2) conditions (4)-(5) are satisfied
for 𝛼 > 1 or condition (10) for 0 < 𝛼 ≤ 1. Then the discrete
spectrum of the operator 𝐿 is real and coincides with the union
of spectra of the self-adjoint operators 𝐿 𝑘 , 𝑘 = 1, 𝑟; that is,
𝑟
𝑥→∞ 𝛾
∞
𝛾0 (𝑥, 𝜆) = 𝑥(𝜆−1)/2 ⋅ exp (−
smallness of perturbations 𝑈𝑘𝑘 (𝑥), and conditions (19), we
conclude that, for every symmetric operator 𝐿󸀠𝑘 , 𝑘 = 1, 𝑟,
there is a case of limit point at infinity. Hence their self-adjoint
extensions 𝐿 𝑘 are the closures of operators 𝐿󸀠𝑘 , respectively.
The operators 𝐿 𝑘 are semibounded below, and their spectra
are discrete.
Let 𝐿 denote the operator extensions 𝐿󸀠 , by requiring that
𝐿 2 (H, (0, ∞)) be the domain of operator 𝐿.
The following theorem is proved in [10].
(21)
Let 𝐿󸀠 denote the minimal differential operator generated
by differential expression 𝑙[𝑦] and the boundary condition
(18), and let 𝐿󸀠𝑘 , 𝑘 = 1, 𝑟, denote the minimal differential
operator on 𝐿 2 (H, (0, ∞)) generated by differential expression 𝑙𝑘 [𝑦𝑘 ] and the boundary conditions (21). Taking into
account the conditions on coefficients, as well as sufficient
−𝑌1∗󸀠󸀠 (𝑥, 𝜆) + 𝑌1∗ (𝑥, 𝜆) ⋅ 𝑉 (𝑥) = 𝜆𝑌1∗ (𝑥, 𝜆) ,
(27)
̃ 𝜆) š 𝑌∗ (𝑥, 𝜆) is a solution to the
the operator function 𝑌(𝑥,
1
left of the equation
̃ 󸀠󸀠 (𝑥, 𝜆) + 𝑌
̃ (𝑥, 𝜆) ⋅ 𝑉 (𝑥) = 𝜆𝑌
̃ (𝑥, 𝜆)
−𝑌
(28)
̃ 𝜆) = cos 𝐴, 𝑌
̃ 󸀠 (0, 𝜆) =
and satisfies the initial conditions 𝑌(0,
sin 𝐴, 𝜆 ∈ C.
4
Abstract and Applied Analysis
Operator solutions of (23) decreasing and increasing
at infinity will be denoted by Φ1 (𝑥, 𝜆), Ψ1 (𝑥, 𝜆), and the
̃ 𝜆) and
corresponding solutions of (28) are denoted by Φ(𝑥,
̃
̃ 𝜆) ∈
Ψ(𝑥, 𝜆). The system operator solutions 𝑌(𝑥, 𝜆), Φ(𝑥,
𝐵(H) of (2) and (28), respectively, will take the form of
̃ 𝑌} = Φ
̃ 󸀠 (𝑥, 𝜆)𝑌(𝑥, 𝜆) − Φ(𝑥,
̃ 𝜆)𝑌󸀠 (𝑥, 𝜆).
Wronskian 𝑊{Φ,
Let us designate
𝐺 (𝑥, 𝑡, 𝜆)
̃ (𝑡, 𝜆)
̃ 𝑌})−1 Φ
0≤𝑥≤𝑡
{𝑌 (𝑥, 𝜆) (𝑊 {Φ,
={
−1
̃
̃
{−Φ (𝑥, 𝜆) (𝑊 {𝑌, Φ}) 𝑌 (𝑡, 𝜆) 𝑥 ≥ 𝑡.
(29)
∞
0
̃ (𝑡, 𝜆) 𝑓 (𝑡) 𝑑𝑡
̃ Φ}) 𝑌
= − ∫ Φ (𝑥, 𝜆) (𝑊 {𝑌,
0
∞
(30)
(33)
+ 𝜒1 (𝑥, 𝜆) − 𝜒2 (𝑥, 𝜆) + 𝜒3 (𝑥, 𝜆) − 𝜒4 (𝑥, 𝜆) ,
̃ Ψ}
̃ Φ})−1 𝑊 {𝑌,
𝜒1 (𝑥, 𝜆) = Φ (𝑥, 𝜆) (𝑊 {𝑌,
̃ (𝑡, 𝜆) 𝑓 (𝑡) 𝑑𝑡,
⋅∫ Φ
𝑎
𝑥
̃ (𝑡, 𝜆) 𝑓 (𝑡) 𝑑𝑡,
𝜒2 (𝑥, 𝜆) = Φ (𝑥, 𝜆) ∫ Ψ
𝑎
̃ 𝑌} (𝑊 {Φ,
̃ 𝑌})
𝜒3 (𝑥, 𝜆) = Φ (𝑥, 𝜆) 𝑊 {Ψ,
−1
(34)
∞
̃ (𝑡, 𝜆) 𝑓 (𝑡) 𝑑𝑡,
⋅∫ Φ
𝑥
∞
̃ (𝑡, 𝜆) 𝑓 (𝑡) 𝑑𝑡.
𝜒4 (𝑥, 𝜆) = Ψ (𝑥, 𝜆) ∫ Φ
𝑥
Let us show that each of these vector-functions 𝜒1 (𝑥, 𝜆),
𝜒2 (𝑥, 𝜆), 𝜒3 (𝑥, 𝜆), and 𝜒4 (𝑥, 𝜆) belongs to 𝐿 2 (H, (0, ∞)).
Since the operator solution Φ(𝑥, 𝜆) decays fairly quickly as
𝑥 → ∞, then |Φ(𝑥, 𝜆)| ∈ 𝐿 2 (0, ∞). It follows that
󵄨
󵄨󵄨
󵄨󵄨𝜒1 (𝑥, 𝜆)󵄨󵄨󵄨 ≤ 𝑐 (𝜆) ⋅ |Φ (𝑥, 𝜆)| ⋅ ∫
𝑎
󵄨
󵄨󵄨 ̃
󵄨󵄨Φ (𝑡, 𝜆)󵄨󵄨󵄨 ⋅ 󵄨󵄨󵄨󵄨𝑓 (𝑡)󵄨󵄨󵄨󵄨 𝑑𝑡
󵄨
󵄨
1/2
𝑥
󵄨
󵄨̃
≤ 𝑐 (𝜆) ⋅ |Φ (𝑥, 𝜆)| ⋅ (∫ 󵄨󵄨󵄨󵄨Φ
(𝑡, 𝜆)󵄨󵄨󵄨󵄨 𝑑𝑡)
𝑎
̃ (𝑡, 𝜆) 𝑓 (𝑡) 𝑑𝑡.
̃ 𝑌}) Φ
+ ∫ 𝑌 (𝑥, 𝜆) (𝑊 {Φ,
𝑥
Theorem 5. The operator 𝑅𝜆 is the resolvent of the operator 𝐿.
4. Proof of Theorem 5
One can directly verify that, for any function 𝑓(𝑥) ∈
𝐿 2 (H, (0, ∞)), the vector-function 𝑦(𝑥, 𝜆) = (𝑅𝜆 𝑓)(𝑥) is a
solution of the equation 𝑙[𝑦] − 𝜆𝑦 = 𝑓 whenever 𝜆 ∉ 𝜎(𝐿).
We will prove that 𝑦(𝑥, 𝜆) ∈ 𝐿 2 (H, (0, ∞)).
Since operator solutions Φ(𝑥, 𝜆) and Ψ(𝑥, 𝜆) form a
fundamental system of solutions of (2), the operator solution
𝑌(𝑥, 𝜆) of (2) satisfying the initial conditions (26) can be
written as 𝑌(𝑥, 𝜆) = Φ(𝑥, 𝜆)𝐴(𝜆)+Ψ(𝑥, 𝜆)𝐵(𝜆), where 𝐴(𝜆) =
̃ 𝑌}, 𝐵(𝜆) = −𝑊{Φ,
̃ 𝑌}; that is,
𝑊{Ψ,
(31)
̃ 𝜆) of (28) can be
Similarly, the operator solution 𝑌(𝑥,
represented in the following form:
̃ (𝑥, 𝜆) = 𝑊 {𝑌,
̃ Φ} Ψ
̃ (𝑥, 𝜆) − 𝑊 {𝑌,
̃ Ψ} Φ
̃ (𝑥, 𝜆) .
𝑌
0
𝑥
−1
̃ 𝑌} − Ψ (𝑥, 𝜆) 𝑊 {Φ,
̃ 𝑌} .
𝑌 (𝑥, 𝜆) = Φ (𝑥, 𝜆) 𝑊 {Ψ,
𝑎
̃ (𝑡, 𝜆) 𝑓 (𝑡) 𝑑𝑡
̃ Φ})−1 𝑌
= − ∫ Φ (𝑥, 𝜆) (𝑊 {𝑌,
𝑥
(𝑅𝜆 𝑓) (𝑥) = ∫ 𝐺 (𝑥, 𝑡, 𝜆) 𝑓 (𝑡) 𝑑𝑡
−1
(𝑅𝜆 𝑓) (𝑥)
where 𝑎 > 0 and
It is proved in [12] that the operator function 𝐺(𝑥, 𝑡, 𝜆)
is Green’s function of the differential operator 𝐿; that is, it
possesses all the classical properties of Green’s function. In
particular, for a fixed 𝑡 the function 𝐺(𝑥, 𝑡, 𝜆) of the variable
𝑥 is an operator solution of (2) on each of the intervals [0, 𝑡),
(𝑡, ∞), and it satisfies the boundary condition (18), and at a
fixed 𝑥, the function 𝐺(𝑥, 𝑡, 𝜆) satisfies (28) in the variable
𝑡 on each of the intervals [0, 𝑥), (𝑥, ∞), and it satisfies the
boundary condition (cos 𝐴)∗ ⋅ 𝑦󸀠 (0) − (sin 𝐴)∗ ⋅ 𝑦(0) = 0.
By definition (28), function 𝐺(𝑥, 𝑡, 𝜆) is meromorphic by
parameter 𝜆 with the poles coinciding with the eigenvalues of
the operator 𝐿.
We consider the operator 𝑅𝜆 defined in 𝐿 2 (H, (0, ∞)) by
the relation
𝑥
By using formulas (31) and (32), we can rewrite relation (30)
as follows:
𝑥
󵄨
󵄨
⋅ (∫ 󵄨󵄨󵄨𝑓 (𝑡)󵄨󵄨󵄨 𝑑𝑡)
𝑎
1/2
(35)
1/2
∞
󵄨
󵄨̃
< 𝑐 (𝜆) ⋅ |Φ (𝑥, 𝜆)| ⋅ (∫ 󵄨󵄨󵄨󵄨Φ
(𝑡, 𝜆)󵄨󵄨󵄨󵄨 𝑑𝑡)
𝑎
∞
󵄨
󵄨
⋅ (∫ 󵄨󵄨󵄨𝑓 (𝑡)󵄨󵄨󵄨 𝑑𝑡)
𝑎
1/2
≤ 𝑐1 (𝜆) ⋅ |Φ (𝑥, 𝜆)| ,
and therefore 𝜒1 (𝑥, 𝜆) ∈ 𝐿 2 (H, (0, ∞)). Similarly we get that
𝜒3 (𝑥, 𝜆) ∈ 𝐿 2 (H, (0, ∞)). First we prove the assertion for
the function 𝜒2 (𝑥, 𝜆), when 𝛼 > 1 and the coefficients of (2)
satisfy the conditions (4)-(5). In this case, we have
𝑥
󵄨󵄨
󵄨̃
󵄨
󵄨
󵄨󵄨
(𝑡, 𝜆)󵄨󵄨󵄨󵄨 󵄨󵄨󵄨𝑓 (𝑡)󵄨󵄨󵄨 𝑑𝑡.
󵄨󵄨𝜒2 (𝑥, 𝜆)󵄨󵄨󵄨 ≤ |Φ (𝑥, 𝜆)| ∫ 󵄨󵄨󵄨󵄨Ψ
𝑎
(36)
By virtue of the asymptotic formulas for the operator solutions Φ(𝑥, 𝜆) and Ψ(𝑥, 𝜆) we obtain that
𝑥
(32)
󵄨󵄨
󵄨
󵄨
󵄨
󵄨󵄨𝜒2 (𝑥, 𝜆)󵄨󵄨󵄨 ≤ 𝑐1 (𝜆) 𝛾0 (𝑥, 𝜆) ∫ 𝛾∞ (𝑡, 𝜆) 󵄨󵄨󵄨𝑓 (𝑡)󵄨󵄨󵄨 𝑑𝑡.
𝑎
(37)
Abstract and Applied Analysis
5
that is, 𝛼 = 1/(2𝑛 − 1) (see (12)). Setting 𝑎(𝛼, 𝜆) = ((1 ⋅ 3 ⋅ . . . ⋅
(2𝑛 − 3))/𝑛!) ⋅ (𝜆/2)𝑛 , we obtain
Let us rewrite this relation in the following form:
󵄨
󵄨󵄨
󵄨󵄨𝜒2 (𝑥, 𝜆)󵄨󵄨󵄨
≤ 𝑐1 (𝜆) 𝛾0 (𝑥, 𝜆) 𝛾∞ (𝑥, 𝜆) ∫
𝑥
𝑎
𝛾∞ (𝑡, 𝜆) 󵄨󵄨
󵄨
󵄨𝑓 (𝑡)󵄨󵄨󵄨 𝑑𝑡.
𝛾∞ (𝑥, 𝜆) 󵄨
(38)
∞ 𝛾
󵄨
󵄨
󵄨󵄨
−𝛼
0 (𝑡, 𝜆) 󵄨󵄨
󵄨󵄨𝜒4 (𝑥, 𝜆)󵄨󵄨󵄨 ≤ 𝑐2 (𝜆) 𝑥 ∫
󵄨󵄨𝑓 (𝑡)󵄨󵄨󵄨 𝑑𝑡 ≤ 𝑐2 (𝜆)
𝑥 𝛾0 (𝑥, 𝜆)
𝑥
By using the definition of the functions 𝛾0 (𝑥, 𝜆) and 𝛾∞ (𝑥, 𝜆)
(see (6)) and by applying the Cauchy- Bunyakovsky inequality we obtain
𝑥
1
V (𝑥)
󵄨 1
󵄨󵄨
(∫ √
󵄨󵄨𝜒2 (𝑥, 𝜆)󵄨󵄨󵄨 ≤ 𝑐1 (𝜆)
2
√V (𝑥) 𝑎 V (𝑡)
⋅ 𝑥−𝛼 (∫ (
𝑎
𝛾0 (𝑡, 𝜆) 2
) 𝑑𝑡)
𝛾0 (𝑥, 𝜆)
𝑥
⋅ exp
(39)
𝑡
∞
󵄨2
󵄨
⋅ (∫ 󵄨󵄨󵄨𝑓 (𝑡)󵄨󵄨󵄨 𝑑𝑡)
0
1/2
∞
󵄨2
󵄨
(∫ 󵄨󵄨󵄨𝑓 (𝑡)󵄨󵄨󵄨 𝑑𝑡)
0
−2𝑥𝛼+1 ((𝑡/𝑥)𝛼+1 − 1)
1+𝛼
1/2
,
(44)
∞
𝑡 2𝑎(𝛼,𝜆)−𝛼
󵄨
󵄨󵄨
−𝛼
󵄨󵄨𝜒4 (𝑥, 𝜆)󵄨󵄨󵄨 ≤ 𝑐3 (𝜆) 𝑥 (∫ ( )
𝑥
𝑥
1/2
⋅ exp (−2 ∫ √V (𝑢)𝑑𝑢) 𝑑𝑡)
1/2
1/2
𝑑𝑡)
.
Replacing variables 𝑡 = 𝑥𝑢, we get
∞
󵄨󵄨
󵄨
−𝛼+1/2
(∫ 𝑢2𝑎(𝛼,𝜆)−𝛼
󵄨󵄨𝜒4 (𝑥, 𝜆)󵄨󵄨󵄨 ≤ 𝑐3 (𝜆) 𝑥
.
1
𝑥
Since 𝑡 ≤ 𝑥, we get exp(−2 ∫𝑡 √V(𝑢)𝑑𝑢) ≤ 1, and then the
latter estimate for 𝜒2 (𝑥, 𝜆) can be rewritten as follows:
𝑥
1
1
󵄨
󵄨󵄨
(∫
𝑑𝑡)
󵄨󵄨𝜒2 (𝑥, 𝜆)󵄨󵄨󵄨 ≤ 𝑐2 (𝜆) 4
√V (𝑥) 𝑎 √V (𝑡)
1/2
∞
1
1
≤ 𝑐2 (𝜆) 4
(∫
𝑑𝑡)
√V (𝑥) 𝑎 √V (𝑡)
(40)
1/2
.
𝑑𝑢)
Since the inequality exp(−𝑥𝛼+1 (𝑢𝛼+1 − 1)/(1 + 𝛼)) ≤ 𝑥−2 holds
for all 𝛼 ∈ (0, 1] and 𝑢 ∈ [1, ∞) with sufficiently large 𝑥, we
have
∞
By formula (4), we get
⋅ exp
(41)
and hence if 𝛼 > 1 and the coefficients of (2) satisfy the
conditions (4) and (5), we have 𝜒2 (𝑥, 𝜆) ∈ 𝐿 2 (H, (0, ∞)). In
the case of V(𝑥) = 𝑥2𝛼 , 0 < 𝛼 ≤ 1, the assertion can be proved
similarly.
For the function 𝜒4 (𝑥, 𝜆) we will conduct the proof for the
case when V(𝑥) = 𝑥2𝛼 , 0 < 𝛼 ≤ 1, and the coefficients of (2)
satisfy condition (10). As in (37) we have
∞
󵄨
󵄨
󵄨
󵄨󵄨
󵄨󵄨𝜒4 (𝑥, 𝜆)󵄨󵄨󵄨 ≤ 𝑐1 (𝜆) 𝛾∞ (𝑥, 𝜆) ∫ 𝛾0 (𝑡, 𝜆) 󵄨󵄨󵄨𝑓 (𝑡)󵄨󵄨󵄨 𝑑𝑡,
𝑥
1+𝛼
(45)
1/2
󵄨
󵄨󵄨
−𝛼−1/2
(∫ 𝑢2𝑎(𝛼,𝜆)−𝛼
󵄨󵄨𝜒4 (𝑥, 𝜆)󵄨󵄨󵄨 ≤ 𝑐3 (𝜆) 𝑥
1
.
𝑐 (𝜆)
󵄨󵄨
󵄨
,
󵄨󵄨𝜒2 (𝑥, 𝜆)󵄨󵄨󵄨 ≤ 4 3
√V (𝑥)
⋅ exp
−2𝑥𝛼+1 (𝑢𝛼+1 − 1)
(42)
−𝑥
𝛼+1
(𝑢
𝛼+1
− 1)
1+𝛼
(46)
1/2
𝑑𝑢)
.
Hence it follows that |𝜒4 (𝑥, 𝜆)| ≤ 𝑐4 (𝛼, 𝜆)𝑥−𝛼−1/2 , and
therefore 𝜒4 (𝑥, 𝜆) ∈ 𝐿 2 (H, (0, ∞)). In case, where 0 < 𝛼 ≤ 1
and (𝛼 + 1)/2𝛼 ∉ 𝑁, and where 𝛼 > 1, the proof is similar.
Thus, 𝑅𝜆 𝑓 ∈ 𝐿 2 (H, (0, ∞)) for any function 𝑓 ∈
𝐿 2 (H, (0, ∞)). This completes the proof.
Since the resolvent 𝑅𝜆 is a meromorphic function of 𝜆, the
poles of which coincide with the eigenvalues of the operator
𝐿, the statement of Theorem 3 can be refined.
Theorem 6. If conditions (4)-(5) where 𝛼 > 1 or condition (10)
where 0 < 𝛼 ≤ 1 is satisfied for (2), then the spectrum of the
operator 𝐿 is real and discrete and coincides with the union of
spectra of self-adjoint operators 𝐿 𝑘 , 𝑘 = 1, 𝑚; that is,
which can be rewritten as follows:
𝑟
𝜎 (𝐿) = ⋃ 𝜎 (𝐿 𝑘 ) .
󵄨󵄨
󵄨
󵄨󵄨𝜒4 (𝑥, 𝜆)󵄨󵄨󵄨
(47)
𝑘=1
≤ 𝑐1 (𝜆) 𝛾0 (𝑥, 𝜆) 𝛾∞ (𝑥, 𝜆) ∫
∞
𝑥
𝛾0 (𝑡, 𝜆) 󵄨󵄨
󵄨
󵄨𝑓 (𝑡)󵄨󵄨󵄨 𝑑𝑡.
𝛾0 (𝑥, 𝜆) 󵄨
(43)
Let us use the asymptotic behavior of the functions 𝛾0 (𝑥, 𝜆)
and 𝛾∞ (𝑥, 𝜆), for example, in the case (𝛼 + 1)/2𝛼 = 𝑛 ∈ 𝑁,
5. Application
Here we consider (2) with matrix coefficients and use the
same notation as in Section 3 (note that could be considered
second-order equation with block-triangular coefficients of
6
Abstract and Applied Analysis
a more general form [14]). Suppose that every symmetric
operator 𝐿󸀠𝑘 is lower semibounded. Let 𝐿 be an arbitrary
extension of the operator 𝐿󸀠 , defined boundary condition at
infinity, and 𝐿 𝑘 an arbitrary self-adjoint extension of the operator 𝐿󸀠𝑘 . If the conditions at infinity determine the Friedrichs
extension 𝐿0𝑘 of the semibounded symmetric operator 𝐿󸀠𝑘 , the
corresponding extension of 𝐿󸀠 will be denoted 𝐿0 . Besides, let
us assume that coefficients of (2) for the problem of semiaxis
are such that discrete spectrum of 𝐿 operator coincides with
the union of discrete spectra of 𝐿 𝑘 operators, 𝑘 = 1, 𝑟,
(sufficient conditions are specified above in Theorem 6).
Denote by nul𝑎 𝑇 the algebraic multiplicity of 0 as an
eigenvalue of 𝑇.
Denote by 𝑁𝑎0 (𝜆) the number of eigenvalues 𝜆0𝑛 < 𝜆 <
𝜆 𝑒 (𝐿0 ) of the operator 𝐿0 counted according to their algebraic
multiplicities. Here 𝜆 𝑒 (𝐿0 ) stands for the lower bound of the
essential spectrum of the operator 𝐿0 .
In [14] is set oscillation theorem of Sturm for equations
with block-triangular matrix potential.
Theorem 7. Suppose the operator 𝐿0 is generated by the
differential expression 𝑙[𝑦] with matrix block-triangular potential, the boundary condition at 0 (18), and such boundary
conditions at the infinity that one gets Friedrichs extensions for
semibounded symmetric operators 𝐿󸀠𝑘 . Then for 𝜆 < 𝜆 𝑒 (𝐿0 ) one
has
∑ nul𝑎 𝑌 (𝑥, 𝜆) = 𝑁𝑎0 (𝜆)
𝑥∈(0,∞)
(48)
(the sum is in those 𝑥 ∈ (0, ∞) for which nul𝑎 𝑌(𝑥, 𝜆) ≠ 0).
In the same article a theorem about the connection
between spectral and oscillation properties for any extension
of the minimal operator is also proved. These theorems are
generalizations for non-self-adjoint operators of the classical
Sturm type oscillation theorems and this problem was considered for the first time.
6. Conclusion
In this work a resolvent is constructed for the SturmLiouville operator with a block-triangular operator potential
increasing at infinite. The structure of the spectrum of such
an operator is obtained.
Competing Interests
The author declared that no competing interests exist.
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Volume 2014
Hindawi Publishing Corporation
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Volume 2014
Discrete Mathematics
Journal of
Volume 2014
Hindawi Publishing Corporation
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Discrete Dynamics in
Nature and Society
Journal of
Function Spaces
Hindawi Publishing Corporation
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Abstract and
Applied Analysis
Volume 2014
Hindawi Publishing Corporation
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Volume 2014
Hindawi Publishing Corporation
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Volume 2014
International Journal of
Journal of
Stochastic Analysis
Optimization
Hindawi Publishing Corporation
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Hindawi Publishing Corporation
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Volume 2014
Volume 2014