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

Abstract

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

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-self-adjoint differential operator, using which the structure of the operator spectrum is set.

2. Preliminary Notes

Let , be finite-dimensional or infinite-dimensional separable Hilbert space with inner product and norm , . Denote . Element will be written in the form , where ,  ,   are identity operators in and accordingly.

We denote by the Hilbert space of vector-valued functions with values in with inner product and the corresponding norm .

Consider the equation with block-triangular operator potentialwhere is a real scalar function, and monotonically, as , and it has monotone absolutely continuous derivative. Also, is a relatively small perturbation; for example, as or . The diagonal blocks ,  , are assumed as bounded self-adjoint operators in ,  .

In case wherewe suppose that coefficients of (2) satisfy relationsLet us consider the functionsIt is easy to see that ,   as . These solutions constitute a fundamental system of solutions of the scalar differential equationwhere is determined by a formula (cf. with the monograph [13])In such a way for all one hasIn case of ,  , we suppose that the coefficients of (2) satisfy the relationNow functions and are defined as follows:These functions also form a fundamental system of solutions of the scalar differential equation, which is obtained by replacing with in formulas (7) and (8).

In [10] the asymptotic behavior of the functions and was established as . If , that is, , then functions and as will have the following asymptotic behavior: In particular, with one has In the case , set , with [] being the integral part of , to obtain the following asymptotic behavior for and at infinity:

In [10] for an equation with matrix coefficients, and in the furtherance for equations with operator coefficients, the following theorem is proved.

Theorem 1. If for (2) conditions (4)-(5) are satisfied for or condition (10) for , then the equation has a unique decreasing at infinity operator solution , satisfying the conditionsAlso, there exists increasing at infinity operator solution , satisfying the conditions

Corollary 2. If , that is, , then, under condition (10), the solutions and have common (known) asymptotic behavior, as in the quality and you can take the following functions:

3. Resolvent of the Non-Self-Adjoint Operator

Let the following boundary condition be given at :where is block-triangular operator of the same structure as the potential (3) of the differential equation (2), and , are the bounded self-adjoint operators in , which satisfy the conditions Together with problem (2) and (18) we consider the separated systemwith the boundary conditions

Let denote the minimal differential operator generated by differential expression and the boundary condition (18), and let ,  , denote the minimal differential operator on 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 ,  , 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 be the domain of operator .

The following theorem is proved in [10].

Theorem 3. Suppose that for (2) conditions (4)-(5) are satisfied for or condition (10) for . Then the discrete spectrum of the operator is real and coincides with the union of spectra of the self-adjoint operators ,  ; 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.

Along with (2) we consider the equation( is adjoint to the operator ). If the space is finite-dimensional, then (23) can be rewritten aswhere and the equation is called the left.

For operator functions letIf is operator solution of (2) and is operator solution of (23), the Wronskian does not depend on .

Now we denote and as the solutions of (2) and (23), respectively, satisfying the initial conditionsBecause the operator function satisfies equationthe operator function is a solution to the left of the equationand satisfies the initial conditions ,  ,  .

Operator solutions of (23) decreasing and increasing at infinity will be denoted by , , and the corresponding solutions of (28) are denoted by and . The system operator solutions of (2) and (28), respectively, will take the form of Wronskian .

Let us designate

It is proved in [12] that the operator function is Green’s function of the differential operator ; that is, it possesses all the classical properties of Green’s function. In particular, for a fixed the function of the variable is an operator solution of (2) on each of the intervals , , and it satisfies the boundary condition (18), and at a fixed , the function satisfies (28) in the variable on each of the intervals , , and it satisfies the boundary condition .

By definition (28), function is meromorphic by parameter with the poles coinciding with the eigenvalues of the operator .

We consider the operator defined in by the relation

Theorem 5. The operator is the resolvent of the operator .

4. Proof of Theorem 5

One can directly verify that, for any function , the vector-function is a solution of the equation whenever . We will prove that .

Since operator solutions and form a fundamental system of solutions of (2), the operator solution of (2) satisfying the initial conditions (26) can be written as , where ; that is,

Similarly, the operator solution of (28) can be represented in the following form:By using formulas (31) and (32), we can rewrite relation (30) as follows:where andLet us show that each of these vector-functions , , , and belongs to . Since the operator solution decays fairly quickly as , then . It follows that and therefore . Similarly we get that . First we prove the assertion for the function , when and the coefficients of (2) satisfy the conditions (4)-(5). In this case, we haveBy virtue of the asymptotic formulas for the operator solutions and we obtain thatLet us rewrite this relation in the following form:By using the definition of the functions and (see (6)) and by applying the Cauchy- Bunyakovsky inequality we obtain Since , we get , and then the latter estimate for can be rewritten as follows:By formula (4), we getand hence if and the coefficients of (2) satisfy the conditions (4) and (5), we have . In the case of ,  , the assertion can be proved similarly.

For the function we will conduct the proof for the case when ,  , and the coefficients of (2) satisfy condition (10). As in (37) we have which can be rewritten as follows:Let us use the asymptotic behavior of the functions and , for example, in the case , that is, (see (12)). Setting , we obtainReplacing variables , we getSince the inequality holds for all and with sufficiently large , we haveHence it follows that , and therefore . In case, where and , and where , the proof is similar.

Thus, for any function . This completes the proof.

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 or condition (10) where 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 , ; that is,

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 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 of the semibounded symmetric operator , the corresponding extension of will be denoted . 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,  , (sufficient conditions are specified above in Theorem 6).

Denote by the algebraic multiplicity of 0 as an eigenvalue of .

Denote by the number of eigenvalues of the operator counted according to their algebraic multiplicities. Here stands for the lower bound of the essential spectrum of the operator .

In [14] is set oscillation theorem of Sturm for equations with block-triangular matrix potential.

Theorem 7. Suppose the operator is generated by the differential expression 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 . Then for one has(the sum is in those for which ).

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 Sturm-Liouville 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.