Abstract and Applied Analysis

Volume 2012 (2012), Article ID 271657, 10 pages

doi:10.1155/2012/271657

Research Article

Characterization of Eigenvalues in Spectral Gap for Singular Differential Operators

Zhaowen Zheng and Wenju Zhang

School of Mathematical Sciences, Qufu Normal University, Shandong, Qufu 273165, China

Received 5 June 2012; Revised 14 August 2012; Accepted 26 August 2012

Academic Editor: Michiel Bertsch

Copyright © 2012 Zhaowen Zheng and Wenju Zhang. 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.

Abstract.

The spectral properties for order differential operators are considered. When given a spectral gap of the minimal operator with deficiency index , arbitrary points in , and a positive integer function such that , has a self-adjoint extension such that each is an eigenvalue of with multiplicity at least .

1. Introduction

In this paper, we consider the following th-order formal symmetric differential expression:

on , where , is a complex-valued -vector function, and are measurable and locally integrable matrices, are Hermitian, and .

The spectral properties of th-order differential expression, particularly the distribution of eigenvalues, have been widely researched in these years (see [1–8] and references cited therein).

Let us recall some known results due to Neumann [1], Stone [2], Friedrichs [3] and Krein [4]. For convenience of the reader in addition to these original sources we will also give text book references [7].

An open interval with is called a spectral gap of a symmetric operator if

and are defined in Section 2. It is easy to find that is a real regular point of . If , we shall also say that is a spectral gap of (the latter definition is not generally used but convenient for our purpose; and it is justified since each interval with is a spectral gap of ). Let be a spectral gap of , then there exists a self-adjoint extension of (for instance the famous Friedrichs extension) such that (here is the regular-form domain of which will be defined in Definition 2.4), this is the reason why we call a spectral gap of .

Suppose in addition that the deficiency index of a symmetric operator is equal to . Let be a self-adjoint extension of . The sum of the multiplicities of the eigenvalues of within the interval is at most , and no point of the continuous spectrum of lies in the interval (cf. [5, Theorem 8.19 and Corollary 2 in Section 8.3]). Conversely, given any finite subset of and positive integers , , such that there exists a self-adjoint extension of such that and the multiplicities of as an eigenvalue of are equal for each (cf. [4]). If one only requires that is some interval within the set of real regular points of , then the corresponding statement is false. For instance, one can give a symmetric operator , such that each is a regular point of , the deficiency index of is equal one and each self-adjoint extension of has a periodic point spectrum with period independent of (cf. [4]).

This paper consists of three sections including the introduction. In Section 2, we present some preliminary materials that include definitions and theorems needed for the rest of the paper. In Section 3, we give three main results in the study of self-adjoint extensions of a minimal operator generated by differential expression . Firstly, we present a partial self-adjointness of the minimal operator . Secondly, if is a spectral gap of , for all is the deficiency index of ) and positive integer function satisfying , has a self-adjoint extension with the following properties:

(i); (ii); (iii).

Finally, given a symmetric operator with the deficiency index being equal to , , being real regular points of , (the interval need not be a spectral gap of and positive integer function satisfying , has a self-adjoint extension with the properties that each is an eigenvalue of with multiplicity at least .

2. Preliminaries

In this section, we introduce notations, definitions, and some theorems that are needed in this paper.

First, we define the following space:

with the inner product

where the weight function is the same as that in (1.1). Denote for . If , then is called square integrable. Here, we note that if is singular, is a quotient space in the sense that if and only if . In this case, is a Hilbert space.

Now we introduce the maximal operator and minimal operator generated by the expression .

Definition 2.1. The maximal operator generated by is defined by

where denotes the collection of functions on which are absolutely continuous locally. Roughly speaking the th quasi-derivative will be collection of terms that, if differentiated times, is “part" of the differential expression (see [7] for details). We know that is densely defined and closed.

Definition 2.2. The preminimal operator generated by is defined by

Obviously, and is Hermitian. It is easy to know that is dense, so is symmetric, and is not closed [8]. The closure of is called the (closed) minimal operator denoted by .

Definition 2.3. Given a linear operator with domain and range in a Hilbert space , the resolvent set of is

the spectrum of is the set . Denote

  ,    ,  .

Obviously, (see [8]).

Definition 2.4. Let be a linear operator in a Hilbert space . The set of regular points of , is called the regular-form domain of , denoted by :

it is an open subset of containing .

Definition 2.5. For a closed operator in a Hilbert space , the essential spectrum of is defined as

We assume that has real regular points, that is, . In this case the deficiency index

does not depend on the special choice of the regular point of and consequently has self-adjoint extensions.

The kernel is the vector space of those solutions of the differential equation which are elements of . Since the space of all solutions of (which in general is not contained in ) has dimension , this implies

Definition 2.6. We say that an operator has pure point spectrum within if .

Proposition 2.7. Let be a self-adjoint operator in a Hilbert space . One has .

Proof. From Definitions 2.3–2.5, we have and , do not belong to , so , is densely defined since it is a self-adjoint operator, so , then we get .

The following results give the relation between deficiency index and self-adjoint extension.

Proposition 2.8. If , , are the preminimal operator, minimal operator and maximal operator generated by , and is one of the self-adjoint extension of , then One has:

Proposition 2.9. If is Hermitian, then (cf. [5, Proposition 2, page 229]).

Proposition 2.10. The deficiency index is constant on each connected subset of . If is Hermitian, then the deficiency index is constant in the upper and lower half-planes (cf. [5, Theorem 8.1]).

Proposition 2.11. A closed symmetric operator is self-adjoint if and only if its deficiency index is equal to .

Proposition 2.12. Let be a symmetric operator,

(a) has self-adjoint extension if and only if its deficiency indices are equal,(b), then A has self-adjoint extensions,(c)if is semibounded, then has self-adjoint extensions,

We can find the proof from Theorem 8.8 in [7].

Proposition 2.13 (see [8]). Let be a closed symmetric operator in and . Set . If there exists a such that , then has a self-adjoint extension and

Furthermore, there exists a self-adjoint extension of such that

3. Main Results

Let , be the minimal operator and maximal operator generated by in (1.1) and let the deficiency index of be equal to . In this section we assume that has real regular points. That is, .

Definition 3.1. A closed subspace of is called a reducing subspace of if , and for all , where denotes the orthogonal projection in onto .

Obviously along with the orthogonal complement of is also a reducing subspace of . It is easy to see that the closed span of an orthogonal system of eigenvectors of is a reducing subspace of . If is a reducing subspace of , then the part of in , that is, the restriction of to may be (and in the following will be) regarded as an operator in the Hilbert space .

Since , there exists a real point of , and the positive and negative deficiency indices of are equal. We choose a one-dimensional subspace of (= ), set

Obviously is a symmetric extension of , and for each regular point of and the corresponding eigenspace. Since the graph of is a one-dimensional extension of the closed graph of , therefore the operator is closed [6].

Theorem 3.2. Assume that the deficiency index of is equal to , let be a spectral gap of , and let be defined as above.

(1)If has an eigenvalue , then is a spectral gap of the restriction of to the Hilbert space .(2)The deficiency index of the restriction of to the Hilbert space is equal to .

Proof. (1) First we consider the case that . Without loss of generality, we may assume that . We have to show that

Assume that for some . Let and . Since reduces ,

for all . Since and span a two-dimensional subspace of and , it follows that for some , this is a contradiction to the hypothesis that is a spectral gap of . This completes the proof in the case that .

The proof in the semibounded case is similar. One only has to replace expression of the form by those of the form , here we omit the details.

(2) Since , that is, , and the restriction of to the Hilbert space is self-adjoint, so the deficiency index of the restriction of onto Hilbert space is equal to .

Theorem 3.3. Suppose the deficiency index of is equal to and is a real regular point of . Let be linearly independent solutions of . One has the following.

(1) has a dimensional symmetric extension with the following properties:

(2). (3)If , then has a self-adjoint extension.

Proof. (1) Since span is a one-dimensional subspace of , we define

Obviously, is a symmetric extension of , , and is an eigenvalue of , and span is the corresponding eigenspace, and is also closed. By induction, we get

and is a closed symmetric operator.

(2) When , then , that is a contradiction, so we have .

(3) When , then , so from Proposition 2.11, we have that is a self-adjoint extension of .

Theorem 3.4. Assume that the deficiency index of is equal to . If is a spectral gap of , then for all (where and positive integer function satisfying , the minimal operator has a self-adjoint extension with the following properties:

(i), (ii)each is aneigenvalue of with multiplicity ,(iii) has pure point spectrum within .

Proof. We choose in , such that    ( denote the cardinality of the set ), in other words each occurs exactly times in the sequence .

We choose a one-dimensional space of and define

Then is a closed symmetric extension of with deficiency index being equal to and is an eigenvalue of with eigensubspace being .

Set and denote by and the restriction of to and , respectively. Obviously is a self-adjoint operator in the one-dimensional Hilbert space with . is a closed symmetric operator in Hilbert space , with deficiency index being equal to , and . Moreover, by Lemma 3.1, is a spectral gap of . Thus we can replace by , by , and by in the above conclusions.

Proceeding further in this way by induction, we obtain sequences and of Hilbert spaces and sequence and of the operators with the following properties:

(i) are closed symmetric extension of which satisfy: , with deficiency indices   , (ii) is a self-adjoint operator on a one-dimensional subspace of , with for each ,(iii) is a closed symmetric operator in the Hilbert space with spectral gap for all .(iv).

Then we have that is a closed symmetric extension of with deficiency index . So is a self-adjoint extension of with the required properties.

Theorem 3.5. Assume the deficiency index of is equal to . Let (where , need not be a spectral gap of ) be real regular points of and let positive integer function satisfy . Moreover, one assumes that has at least square integrable solutions. Then has a self-adjoint extension such that each is an eigenvalue of with multiplicity at least equal to .

Proof. Choose a sequence in such that each occurs at least times in this sequence (the times of the appears in the sequence satisfies , is the number of square integrable solutions of ; the which is not in will be selected as a regular point of satisfying ). Since the deficiency index of is equal to , the negative and positive deficiency indices are equal to , so we select a normalized orthogonal basis and . Now, we define by induction, orthonormal sequences as follows:

for , we choose normalized element with

Suppose for given , the () are well defined with the properties

(those are eigenfunctions of relating to , so they are orthogonal if , and if , we select different orthogonal eigenfunctions of relating to ), we choose a normalized element with

When , we stop the induction process. So is an orthonormal sequence with the following properties:

(i) for all ,(ii) and for all ,(iii) for all .

Set

Obviously is a symmetric extension of and for , so each is an eigenvalue of with multiplicity at least equal to . For every , we have

It follows that for all . Thus we have

Analogously, it can be shown that

Thus and consequently is a self-adjoint extensions which has the properties mentioned above.

Acknowledgments

The authors sincerely thank the referee for the constructive suggestions and corrections. This paper was partially supported by the NNSF of China (Grant 11171178, 11271225) and NSF of Shandong (Grant ZR2009AQ010).

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