Relationships among Three Multiplicities of a Differential Operator’s Eigenvalue ()

Shouzhong Fu, Zhong Wang

School of Mathematics and Statistics, Zhaoqing University, Guangdong, China.

**DOI: **10.4236/am.2014.515211
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School of Mathematics and Statistics, Zhaoqing University, Guangdong, China.

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.

Keywords

Differential Operators, Eigenvalue, Algebraic Multiplicities, Geometric Multiplicities, Analytic Multiplicities

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Fu, S. and Wang, Z. (2014) Relationships among Three Multiplicities of a Differential Operator’s Eigenvalue. *Applied Mathematics*, **5**, 2185-2194. doi: 10.4236/am.2014.515211.

1. Introduction

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 [1] -[3] , while the singular case started with the celebrated work of Weyl in 1910 introducing the limit-point (LP) and limit-circle (LC) dichotomy [4] . Another important milestone in this area is the Glazman-Krein-Naimark (GKN) theorem [5] in 1950, see also [6] for generalizations (which will be included in the theorem). This theorem gives a one-to-one correspondence between the self-adjoint differential operators in a Hilbert function space representing a given QDE and the unitary isometries on an appropriate finite-dimensional subspace (or equivalently, certain Lagrange subspaces of some finite dimensional quotient space, see [7] ). In both regular case and singular case, the GKN theorem also yields a characterization of the self-adjoint operators in terms of linear complex boundary conditions (BC). So, the spectral problem of a linear ordinary differential equation (QDE) with boundary conditions maybe turn to study it of a linear ordinary differential operator [5] [8] [9] . It is well-known that the investigation of different self-adjoint extensions of symmetric operators and the estimation of the location and multiplicity of their point spectra are among fundamental mathematical problems arising in any quantum mechanical model ([10] , Chapter VIII, Section 11). While the eigenvalues of a linear ordinary differential operator were studied, three kinds of multiplicity (analytic multiplicity, geometric multiplicity and algebraic multiplicity) of an eigenvalue were defined and accompanied. Three kinds of multiplicity are often confused in some paper, and a problem how about the relationships among three kinds of multiplicity of an eigenvalue has arisen.

The differential equation

(1.1)

with boundary conditions

(1.2)

will be studied in present paper, where and a.e. on, the functions are real-valued, measurable over and Lebesgue integrable on all compact subset of.

The endpoint is said to be regular if and each of the functions is integrable in every interval; otherwise is said to be singular. Similar definitions apply to endpoint. The differential expression is said to be regular if it is regular at both endpoints, and otherwise is said to be singular.

We assume throughout that (1.1) is regular, and the functions are sufficiently smooth and Lebesgue integrable on, then the boundary conditions may be written as:

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

(1.3)

and, where and which come from the coefficient matrix

of the boundary conditions are complex matrices, while is a fixed matrix for Lagrange bilinear form of differential expression in (1.1).

Let, and be the transpose of, then, for

(the domain of maximal operator generated by),

and following results are true (see [5] [6] [8] ):

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, called the characteristic function of the problem. The analytic multiplicity of an eigenvalue is the order of the eigenvalue as a zero of, the geometric multiplicity of an eigenvalue is the number of linearly independent eigenfunctions for the eigenvalue, and the algebraic multiplicity of an eigenvalue is the dimension of its root subspace (a subspace is spanned by the eigenvectors and its associated vectors).

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, [5] [11] -[14] ). The geometric multiplicity is always defined and is more widely used in spectral theory (see [4] [5] [8] -[10] [15] ). However, the algebraic multiplicities and associated vectors (functions) of an eigenvector (eigenfunction) are used for the completeness of eigenvectors (eigenfunctions) of non self-adjoint operators (see [15] - [17] ). Therefore, it is of fundamental interest to compare the three multiplicities of an eigenvalue for differential operators.

Naimark studied the relationship between the algebraic and analytic multiplicities of an eigenvalue of highorder linear differential operators in [5] , and obtained the equivalence of the algebraic and analytic multiplicities of an eigenvalue of high-order linear differential Equation (1.1) with linear boundary conditions (1.2). From then, the relationships among the three multiplicities have been payed a good deal of attentions, and have had a strong appeal to studying.

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 ([5] [11] -[14] [18] -[22] ). Sturm-Liouville problems (SLP) are differential equation:

(1.4)

with boundary conditions

(1.5)

where a.e. on, , and are matrices,.

For the regular SLP, i.e. both endpoint and are regular, the problem (1.4)-(1.5) is a self-adjiont SLP if the coefficients matrixes in (1.5) satisfy (1.3) and.

The equality of the analytic and geometric multiplicities in the case of separated boundary conditions (BC) was proved in [13] , while the case of coupled BC’s was settled in [18] . The equality of the two multiplicities in the case of singular self-adjoint SLP with LC non-oscillatory end points was shown in [19] using a regularization; the equality in the case of all singular self-adjoint SLP with LC end points was recently established in [21] , based on the equality of the regular self-adjoint SLP and certain regular approximations.

The proof in [18] uses some sophisticated identities involving (the function in [18] differs from by a constant) and certain values of a fundamental set of solutions of (1.4). It seems to us that it is very hard to find similar identities for a higher order QDE by this way.

The basic idea of proof in [19] is as follows: for any eigenvalue of geometric multiplicity 1, they can give a smooth curve in the space of self-adjoint BC through the BC involved such that the composition of with a continuous eigenvalue branch through has a non-zero derivative along the curve at, which then implies that has a non-zero derivative at. Here, a continuous eigenvalue branch through means a continuous function defined on a neighborhood O of A in such that its value at A equals and its value at each BC is an eigenvalue for.

In [22] , we generalized the proof in [19] , and gave a new and unified proof of the equality between the analytic and geometric multiplicities of any eigenvalue of self-adjoint SLP in the regular case based on the geometric classifications of self-adjoint BC.

The classifications of self-adjoint BC about higher order differential operator are more complicated in geometric [7] , and it seems rather complicated to find similar identities for a higher order QDE by the same method in [22] . Recently, this result was generalized to the higher order differential equations with self-adjoint boundary conditions in [23] , but the proof was not easy.

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, [7] or [24] ).

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.

2. Geometric and Algebraic Multiplicities

The definitions of the geometric and algebraic multiplicities for an eigenvalue of a linear operator are from [15] . Recall that a complex number is called an eigenvalue of linear operator if there exists a non-zero element such that; in this case, is called an eigenfunction of for. The eigenfunctions for span a subspace of, , called the eigenspace for; and the geometric multiplicity of is the dimension of its eigenspace, denoted by, i.e.

A non-zero element is called a root vector of for a complex number if for some. In this case, must be an eigenvalue. Together with the vector, the root vectors of span a linear subspace of, , called the root lineal for; and the algebraic multiplicity of is the dimension of its root lineal, denoted by, i.e.

If an element is not an eigenvector for, then it is a root vector for if and only if there is a such that is an eigenvector for provided and for. A root vector is called an associated vector (or adjoint vector) if it is not an eigenvector. The theory of associated functions (vectors) of differential operator was originated by Keldysh [16] .

In general, the system of eigenvectors and associated vectors of is not complete in Hilbert space. From the definition of the algebraic multiplicities and geometric multiplicity of an eigenvalue of linear operator T in H, the eigenvectors of belong to root lineal of (i.e. ), so, we have the following result:

Theorem 2.1. The geometric multiplicity of any eigenvalue of linear operator in Hilbert space does not exceed its algebraic multiplicity. i.e.

(2.1)

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 in,

(2.2)

are eigenvalues of operator, is an eigenvector for the eigenvalue, and the eigenspace. So, the geometric multiplicity of the eigenvalue is equal to 1, i.e.. But, have solutions, where, , the associated vector of is or, and the root lineal or. Thus, the algebraic multiplicity of eigenvalue is equal to 2, i.e..

Example 2.3. We consider the differential equation

(2.3)

with boundary conditions

(2.4)

in Hilbert space. After simple calculation, is an eigenvalue of boundary problem (1.3)-(1.4), is corresponding eigenvector, and the eigenspace of is. So, the geometric multiplicity of eigenvalue is equal to 1, i.e.. But, have solutions, where

, but has not any solutions, the associated vector of is and the root lineal. Thus, the algebraic multiplicity of eigenvalue is equal to 2, i.e..

If is a spacial operator in Hilbert space, then the geometric multiplicity of any eigenvalues of linear operator maybe equal to its algebraic multiplicity, such as:

Theorem 2.4. If is a self-adjoint operator in Hilbert space, then the geometric multiplicity of any eigenvalues of linear operator is equal to its algebraic multiplicity.

Proof: We only need to prove that the eigenspace for an eigenvalue of operator is equal to the root lineal of.

From the definitions of eigenspace and root lineal of the eigenvalue, we have

where and represent the eigenspace and the root lineal of, is an (isolated) eigenvalue of, i.e..

If , then there exist a associated function (,) and an eigenfunction (,), such that, and.

From the last equation, we have

(2.5)

By self-adjointness of operator, the eigenvalue is a real number, and for any. Then,

(2.6)

and there is a contradictory to the fact. Therefore,

(2.7)

and the proof is complete. □

3. Analytic and Algebraic Multiplicities

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 be the fundamental solution of (1.1) satisfying

(3.1)

and denote the Wronskian of (1.1) with respect to , then. The determinant of matrix is denoted by, i.e.

(3.2)

Theorem 3.1. The is an entire function of, and a complex number is an eigenvalue of the boundary value problem consisting of (1.1) and (1.2) if and only if

(3.3)

Proof: Simply calculate or see [8] and [5] .

The entire function, unique up to a non-zero constant multiple, is called the characteristic function for the boundary value problem consisting of (1.1) and (1.2). The analytic multiplicity of an isolated eigenvalue is the order of the eigenvalue as a zero of, denoted by.

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 function for the boundary value problem (2.3) and (2.4) can be easily obtained after calculating.

(3.4)

is a 3-order zero of the, so, the analytic multiplicity of the eigenvalue is 3.

With the results in Example 2.3, we have proven that the geometric multiplicity, algebraic multiplicity and analytic multiplicity of the eigenvalue of boundary value problem (1.3)-(1.4) are 1, 2 and 3, and for the eigenvalue of boundary value problem (1.3)-(1.4).

But the linear differential Equation (1.1) with linear boundary conditions (1.2), Naimark had the following theorem in [5] (Chapter:).

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.

(3.5)

The algebraic multiplicity of an eigenvalue of self-adjoint SLP factually is the analytic multiplicity in [11] - [14] , because the authors realized that the equivalence of the algebraic and analytic multiplicities for any eigenvalues of self-adjoint SLP is a foundational fact.

4. Equalities among Three Multiplicities of an Eigenvalue of Self-Adjoint Differential Operators

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, we use to denote the vector space of matrices with complex entries and its open subset consisting of the elements with the maximum rank. and are defined similarly. When a capital Latin or Greek letter stands for a matrix, the entries of the matrix will be denoted by the corresponding lower case letter with two indices. If, then and are the transpose and the complex conjugate transpose of, respectively. The general linear group is a complex Lie group under the matrix multiplication, while the special linear group consists of all elements of with determinant 1 and is a Lie subgroup of. and are defined similarly. Let be an open interval, bounded or unbounded. Assume that is one of the spaces, , and. We denote by the space of Lebesgue integrable S-valued functions on, and the space of S-valued functions which are absolutely continuous on all compact subintervals of.

For the rest of this paper, we use to denote a fixed number satisfying, and with

. When is even and is finite interval, there is a special case of the QDE (1.1). This special case was studied by Naimark [5] and Weidman [9] .

For any and a.e. on, are integrable over

, and are times continuous differentiable function on,. We define

(4.1)

then also belongs to. Let, the quasi-derivatives of associated with are given by

and

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

(4.2)

Thus, and can be used as the coefficient matrix of (4.2), (4.2) is equivalent to its matrix form

(4.3)

The quasi-differential expression in associated with for or or are of special interest:

while, in general, is equal to

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

(4.4)

where such that. Note that equivalent linear algebraic equations of the form (1.2) define the same BC. Each value of for which the QDE (4.2) has a nontrivial solution satisfying the BC (1.4) is called an eigenvalue of the BVP consisting of (4.2) and (4.4), and a solution to this problem is called an eigenfunction for this eigenvalue.

Since (4.2) has exactly linearly independent solutions, the geometric multiplicity of any eigenvalue is an integer not smaller than 1 and not greater than.

Theorem 4.1. A number is an eigenvalue of the boundary value problem consisting of (4.2) and (4.4) if and only if

(4.5)

where is the fundamental solution of (1.3) satisfying, is the identity matrix.

Proof: Simply calculate or see [7] . □

Theorem 4.2.

Proof: From the definition of quasi-derives of associated with, we have, where, , and

(4.6)

is a inverse matrix function on because of over. Let denote the Wronskian of (1.1) with respect to, , then , and. Thus, and. So, the equality in this theorem was proved.

A coefficient matrix is said to be symmetric if

(4.7)

where

(4.8)

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

(4.9)

Theorem 4.3. The differential operator associated with (1.1) and (1.2) (or (4.2) and (4.4)) is a self-adjoint operator if the boundary conditions (1.2) (or (4.4)) satisfy (1.3) and (or is self-adjoint).

Proof: See [7] . □

Theorem 4.4. The analytic, algebraic and geometric multiplicity of any eigenvalue for a self-adjoint differential 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 and satisfy (4.9)), and equal to the multiplicities of the eigenvalue as a zero point of in (3.3) (or in (4.5)).

Example 4.6. We consider the differential equation

(4.10)

with boundary conditions

(4.11)

in Hilbert space. After simple calculation, are eigenvalues of boundary problem (4.10)-(4.11), and are corresponding eigenvectors for, and the eigenspace of is. So, the geometric multiplicity of eigenvalue is equal to 2, i.e.. Boundary problem (4.10)-(4.11) is a self-adjoint problem because of boundary conditions (4.11) satisfying (4.9), so, the algebraic multiplicity of eigenvalue is also equal to 2, i.e.. We also have that

(4.12)

are 2-order zero of the, so, the analytic multiplicity of the eigenvalue is 2, i.e.. Thus, the analytic, algebraic and geometric multiplicity of any eigenvalue for boundary problem (4.10)-(4.11) are equal to 2, i.e..

Funding

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

NOTES

^{*}Corresponding author.

Conflicts of Interest

The authors declare no conflicts of interest.

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