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In this paper, the algebraic, geometric and analytic multiplicities of an eigenvalue for linear differential operators are defined and classified. The relationships among three multiplicities of an eigenvalue of the linear differential operator are given, and a fundamental fact that the algebraic, geometric and analytic multiplicities for any eigenvalue of self-adjoint differential operators are equal is proven.

The study of spectral problems for linear ordinary differential equations (more generally, quasi-differential equations, to be abbreviated as QDE) originated from a series of seminal papers of Sturm and Liouville in [

The differential equation

with boundary conditions

will be studied in present paper, where

The endpoint

We assume throughout that (1.1) is regular, and the functions

In the regular case, the GKN characterization of self-adjointness in terms of the complex boundary conditions can be simply expressed as the algebraic equation

and

Let

and following results are true (see [

1)

2)

It is well-known that the spectrum of such a problem consists of an infinite number of real eigenvalues and has no finite accumulation point. The eigenvalues are precisely the zeros of an entire function

The analytic multiplicity of an eigenvalue gives the maximum number of new eigenvalues into which the original eigenvalue can split when the spectral problem involved. So, it is natural to use the analytic multiplicity to count eigenvalues, and the analytic multiplicity plays an important role in the study of the dependence of the eigenvalues of a spectral problem on the differential equation boundary value problem (see, for example, [

Naimark studied the relationship between the algebraic and analytic multiplicities of an eigenvalue of high- order linear differential operators in [

Over the last decades, the fact that the analytic and geometric multiplicities of an eigenvalue of self-adjoint Sturm-Liouville problems are equal has been solved ([

with boundary conditions

where

For the regular SLP, i.e. both endpoint

The equality of the analytic and geometric multiplicities in the case of separated boundary conditions (BC) was proved in [

The proof in [

The basic idea of proof in [

In [

The classifications of self-adjoint BC about higher order differential operator are more complicated in geometric [

In order to classify three multiplicities of an eigenvalue for linear differential operators, to obtain the relationships among three multiplicities, and to have a short and non-technical presentation so that the main idea of the general proof can be made transparent, we only give the general proof for regular self-adjoint QDE in this paper. For arbitrary self-adjoint nth-order QDE in singular end points with defect index n, the proof is basically the same (with only obvious minor changes), but the introduction of the self-adjoint BC and the definition of the characteristic function are more involved (see, for example, [

It is the main purpose, therefore, in the present work, to give the definitions of three kinds of multiplicities of an eigenvalue for linear differential operators and the relationships among them. In Section 2, we give the definitions of the geometric and algebraic multiplicities and the relationship between them. The definition of the analytic multiplicity for an eigenvalue of linear differential operators and the relationship between its analytic and algebraic multiplicities is given in Section 3. In last section, we have the equalities among three multiplicities of an eigenvalue for a self-adjoint linear differential operator.

The definitions of the geometric and algebraic multiplicities for an eigenvalue of a linear operator are from [

A non-zero element

If an element

In general, the system of eigenvectors and associated vectors of

Theorem 2.1. The geometric multiplicity of any eigenvalue of linear operator

In general, the algebraic multiplicity is grater than the geometric multiplicity of an eigenvalue of operators. For example.

Example 2.2. We consider an operator

Example 2.3. We consider the differential equation

with boundary conditions

in Hilbert space

If

Theorem 2.4. If

Proof: We only need to prove that the eigenspace for an eigenvalue

From the definitions of eigenspace and root lineal of the eigenvalue

where

If

From the last equation, we have

By self-adjointness of operator

and there is a contradictory to the fact

and the proof is complete. □

We also introduce some notations here and review some basic facts about the problem of differential Equation (1.1) with boundary conditions (1.2). Let

and

Theorem 3.1. The

Proof: Simply calculate or see [

The entire function

From the definition of analytic multiplicity, only the eigenvalues of boundary value problems have analytic multiplicity. In general, the analytic multiplicity isn’t equal to the other multiplicities for an eigenvalue of some boundary value problems.

Example 3.2. We still study the boundary value problem (2.3)-(2.4) in Example 2.3. The characteristic func- tion

With the results in Example 2.3, we have proven that the geometric multiplicity, algebraic multiplicity and analytic multiplicity of the eigenvalue

But the linear differential Equation (1.1) with linear boundary conditions (1.2), Naimark had the following theorem in [

Theorem 3.3. The analytic multiplicity of any eigenvalues of the boundary value problem consisting of (1.1) and (1.2) is equal to its algebraic multiplicity. i.e.

The algebraic multiplicity of an eigenvalue of self-adjoint SLP factually is the analytic multiplicity in [

In this section, we first collect some basic statements about higher order differential operator (especially, the self-adjoint differential operator with high-order), and then prove the equalities among analytic, algebraic and geometric multiplicities.

For any

For the rest of this paper, we use

For any

then

and

then, ordinary differential Equation (1.1) is equivalent quasi differential equation (QDE)

Thus,

The quasi-differential expression in

while, in general,

We now turn to the BVP consisting of the general QDE (1.2) and a (linear two-point) BC defined by

where

Since (4.2) has exactly

Theorem 4.1. A number

where

Proof: Simply calculate or see [

Theorem 4.2.

Proof: From the definition of quasi-derives of

is a inverse matrix function on

A coefficient matrix

where

The boundary condition (1.4) is said to be self-adjoint if

Theorem 4.3. The differential operator

Proof: See [

Theorem 4.4. The analytic, algebraic and geometric multiplicity of any eigenvalue for a self-adjoint differ- ential operator associated with (1.1) and (1.2) (or (4.2) and (4.4)) are equal.

Proof: From Theorem 2.4 and Theorem 3.3, we get the conclusion immediately. □

Corollary 4.5. The analytic, algebraic and geometric multiplicity of any eigenvalue of the differential operator L associated with (1.1) and (1.2) (or (4.2) and (4.4)) are equality when the coefficient matrixes A and B satisfy (1.3) (or

Example 4.6. We consider the differential equation

with boundary conditions

in Hilbert space

Work partially supported by the National Nature Science Foundation (11171295).

^{*}Corresponding author.