Resolvent for Non-Self-Adjoint Differential Operator with Block-Triangular Operator Potential

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 x→∞ Φ (x, λ) γ0 (x, λ) = I, lim x→∞ Φ󸀠 (x, λ) γ󸀠 0 (x, λ) = I. (15) Also, there exists increasing at infinity operator solution Ψ(x, λ), satisfying the conditions lim x→∞ Ψ (x, λ) γ∞ (x, λ) = I, lim x→∞ Ψ󸀠 (x, λ) γ󸀠 ∞ (x, λ) = I. (16) 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: γ0 (x, λ) = x(λ−1)/2 ⋅ exp(−x2 2 ) , γ∞ (x, λ) = x−(λ+1)/2 ⋅ exp(x2 2 ) . (17) 3. Resolvent of the Non-Self-Adjoint Operator Let the following boundary condition be given at x = 0: cosA ⋅ y󸀠 (0) − sinA ⋅ 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 Akk, k = 1, r, are the bounded self-adjoint operators in Hk, which satisfy the conditions −π2 Ik ≪ Akk ≤ π2 Ik. (19) Together with problem (2) and (18) we consider the separated system lk [yk] = −y󸀠󸀠 k + (V (x) Ik + Ukk (x)) yk = λyk, k = 1, r (20) with the boundary conditions cosAkk ⋅ y󸀠 k (0) − sinAkk ⋅ yk (0) = 0, k = 1, r. (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 L2(H, (0,∞)) generated by differential expression lk[yk] and the boundary conditions (21). Taking into account the conditions on coefficients, as well as sufficient 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 Lk are the closures of operators L󸀠k, respectively. The operators Lk are semibounded below, and their spectra are discrete. Let L denote the operator extensions L󸀠, by requiring that L2(H, (0,∞)) be the domain of operator L. The following theorem is proved in [10]. Theorem3. 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 Lk, k = 1, r; that is, σd (L) = r ⋃


Introduction
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 non-Hermitian 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.
We denote by  2 (H, (0, ∞)) the Hilbert space of vectorvalued functions () with values in H with inner product and the corresponding norm ‖ ⋅ ‖.
Consider the equation with block-triangular operator potential where V() is a real scalar function, and 0 < V() → ∞ monotonically, as  → ∞, and it has monotone absolutely continuous derivative.Also, () is a relatively small perturbation; for example, The diagonal blocks   (),  = 1, , are assumed as bounded self-adjoint operators in   ,   :   →   .
In case where we suppose that coefficients of (2) satisfy relations Let us consider the functions It is easy to see that  0 () → 0,  ∞ () → ∞ as  → ∞.These solutions constitute a fundamental system of solutions of the scalar differential equation where () is determined by a formula (cf.with the monograph [13]) In such a way for all  ∈ [0, ∞) one has In case of V() =  2 , 0 <  ≤ 1, we suppose that the coefficients of (2) satisfy the relation Now functions  0 (, ) and  ∞ (, ) are defined as follows: These functions also form a fundamental system of solutions of the scalar differential equation, which is obtained by replacing V() with V() −  in formulas ( 7) and (8).

Resolvent of the Non-Self-Adjoint Operator
Let the following boundary condition be given at  = 0: where  is block-triangular operator of the same structure as the potential () (3) of the differential equation ( 2), and   ,  = 1, , are the bounded self-adjoint operators in   , which satisfy the conditions Together with problem (2) and (18) we consider the separated system with the conditions Let   denote the minimal differential operator generated by differential expression [] and the boundary condition (18), and let    ,  = 1, , denote the minimal differential operator on  2 (H, (0, ∞)) generated by differential expression   [  ] and the boundary conditions (21).Taking into account the conditions on coefficients, as well as sufficient smallness of perturbations   (), and conditions (19), we conclude that, for every symmetric operator    ,  = 1, , there is a case of limit point at infinity.Hence their self-adjoint extensions   are the closures of operators    , respectively.The operators   are semibounded below, and their spectra are discrete.
Let  denote the operator extensions   , by requiring that  2 (H, (0, ∞)) be the domain of operator .
The following theorem is proved in [10].
Theorem 3. Suppose that for ( 2) conditions ( 4)-( 5) are satisfied for  > 1 or condition (10) for 0 <  ≤ 1.Then the discrete spectrum of the operator  is real and coincides with the union of spectra of the self-adjoint operators   ,  = 1, ; that is, Comment 4. Note that this theorem contains a statement of the discrete spectrum of the non-self-adjoint operator  only and no allegations of its continuous and residual spectrum.
By definition (28), function (, , ) is meromorphic by parameter  with the poles coinciding with the eigenvalues of the operator .
We consider the operator   defined in  2 (H, (0, ∞)) by the relation
Since the resolvent   is a meromorphic function of , the poles of which coincide with the eigenvalues of the operator , the statement of Theorem 3 can be refined.Theorem 6.If conditions ( 4)- (5) where  > 1 or condition (10) where 0 <  ≤ 1 is satisfied for (2), then the spectrum of the operator  is real and discrete and coincides with the union of spectra of self-adjoint operators   ,  = 1, ; that is, (47)

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 a more general form [14]). Suppose that every symmetric operator    is lower semibounded.Let  be an arbitrary extension of the operator   , defined boundary condition at infinity, and   an arbitrary self-adjoint extension of the operator    .If the conditions at infinity determine the Friedrichs extension  0  of the semibounded symmetric operator    , the corresponding extension of   will be denoted  0 .Besides, let us assume that coefficients of (2) for the problem of semiaxis are such that discrete spectrum of  operator coincides with the union of discrete spectra of   operators,  = 1, , (sufficient conditions are specified above in Theorem 6).
Denote by nul   the algebraic multiplicity of 0 as an eigenvalue of .
Denote by  0  () the number of eigenvalues  0  <  <   ( 0 ) of the operator  0 counted according to their algebraic multiplicities.Here   ( 0 ) stands for the lower bound of the essential spectrum of the operator  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.

Conclusion
In this work a resolvent is constructed for the Sturm-Liouville operator with a block-triangular operator potential increasing at infinite.The structure of the spectrum of such an operator is obtained.