Relationships among Three Multiplicities of a Differential Operator’s Eigenvalue

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.


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 with boundary conditions will be studied in present paper, where ( ) and ( ) 0 w x > a.e. on ( ) , a b , the functions < ; otherwise a is said to be singular.Similar definitions apply to endpoint b .The differential expression ( ) l y 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 ( ) , a b , 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 ( ) and following results are true (see [5] [6] [8]): 1) 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 ( ) λ ∆ , called the characteristic function of the problem.The analytic multiplicity of an eigenvalue is the order of the eigenvalue as a zero of ( ) λ ∆ , 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: with boundary conditions where 0 For the regular SLP, i.e. both endpoint a and b are regular ( ) 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 ( ) λ ∆ (the function ( ) D λ in [18] differs from ( ) λ −∆ 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 * λ of geometric multiplicity 1, they can give a smooth curve in the space C B of self-adjoint BC through the BC A involved such that the composi- tion of ( ) λ ∆ with a continuous eigenvalue branch through * λ has a non-zero derivative along the curve at A , which then implies that ( ) λ ∆ has a non-zero derivative at * λ .Here, a continuous eigenvalue branch through * λ means a continuous function defined on a neighborhood O of A in C B such that its value at A equals * λ and its value at each BC O ∈ B is an eigenvalue for B .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.

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 0 λ is called an eigenvalue of linear operator T if there exists a non-zero ele- ( ) ( ) is not an eigenvector for 0 λ , then it is a root vector for 0 λ if and only if there is a =  .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 T is not complete in Hilbert space H . From the definition of the algebraic multiplicities and geometric multiplicity of an eigenvalue of linear operator T in H, the eigenvectors of 0 λ belong to root lineal of 0 λ (i.e. ( ) ( ) ), so, we have the following result: Theorem 2.1.The geometric multiplicity of any eigenvalue of linear operator T in Hilbert space does not exceed its algebraic multiplicity.i.e.

( ) ( )
In general, the algebraic multiplicity is grater than the geometric multiplicity of an eigenvalue of operators.For example. .So, the geometric multiplicity of eigenvalue 2 λ = π is equal to 1, i.e. .Thus, the algebraic multiplicity of eigenvalue 2 λ = π is equal to 2, i.e.
If T is a spacial operator in Hilbert space H , then the geometric multiplicity of any eigenvalues of linear operator T maybe equal to its algebraic multiplicity, such as: Theorem 2.4.If T is a self-adjoint operator in Hilbert space H , then the geometric multiplicity of any eigenvalues of linear operator T is equal to its algebraic multiplicity.
Proof: We only need to prove that the eigenspace for an eigenvalue λ of operator T is equal to the root lineal of λ .
From the definitions of eigenspace and root lineal of the eigenvalue λ , we have ( ) , then there exist a associated function 1 f ( ( ) ( ) and an eigenfunction f ( ( ) From the last equation, we have By self-adjointness of operator T , the eigenvalue λ is a real number, and ( ) ( ) for any ( ) and there is a contradictory to the fact ( ) 0 and the proof is complete.□

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 , , , , , , n y x y x y x λ λ λ  be the fundamental solution of (1.1) satisfying and , , , , , , , , W a λ = .The determinant of matrix ( ) ( ) .
The entire function ( ) λ ∆ , 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 ( ) But the linear differential Equation (1.1) with linear boundary conditions (1.2), Naimark had the following theorem in [5] (Chapter: 18 P ).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 [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.

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 the rest of this paper, we use ( ) n ∈  to denote a fixed number satisfying 2 n ≥ , and ( ) When n is even ( ) and J 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 .We define , , , , , the quasi-derivatives of y associated with P are given by [ ] ( ) for 0,1, , 1, Thus, P and W can be used the coefficient matrix of (4.2), (4.2) is equivalent to its matrix form The quasi-differential expression in y associated with Q for 2 n = or 4 n = or 6 n = are of special interest: We now turn to the BVP consisting of the general QDE (1.2) and a (linear two-point) BC defined by of maximal operator generated by ( ) l y ), case, 0 y is called an eigenfunction of T for 0 λ .The eigen- functions for 0 λ span a subspace of ( ) n N ∈ .In this case, 0 λ must be an eigenvalue.Together with the vector 0 φ = , the root vectors of 0 λ span a linear subspace of ( ) root lineal for 0 λ ; and the algebraic multiplicity of 0 λ is the dimension of its root lineal, denoted by

1 n 1 L=
λ = .But, ( ) has not any solutions, the associated vector of y is 1 y and the root lineal of λ , λ is an (isolated) eigenvalue of T , i.e.

2 ) 3 . 1 .
Theorem The ( ) λ ∆ is an entire function of λ , and a complex number λ is an eigenvalue of the boundary value problem consisting of (1.1) and (1.2) if and only if

3 . 2 .the analytic multiplicity of the eigenvalue 2 λ = π is 3 .
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 We still study the boundary value problem (2.3)-(2.4) in Example 2.3.The characteristic function ( ) λ ∆ for the boundary value problem (2.3) and (2.4) can be easily obtained after calculating.With the results in Example 2.3, we have proven that the geometric multiplicity, algebraic multiplicity and analytic multiplicity of the eigenvalue 2 λ = π of boundary value problem (1.3)-(1.4)are 1, 2 and 3of boundary value problem (1.3)-(1.4).

A,
subset consisting of the elements with the maximum rank 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 are the transpose and the complex conjugate transpose of A , respectively.The general linear group complex Lie group under the matrix multiplication, while the special linear group ( ) , SL n  consists of all elements of ( ) , GL n  with determinant 1 and is a Lie subgroup of SL n  are defined similarly.Let J be an open interval, bounded or unbounded.Assume that S is one of the spaces n  , n  , denote by ( ) , L J S the space of Lebesgue integrable S-valued functions on J , and ( ) , loc AC J S the space of S-valued functions which are absolutely continuous on all compact subintervals of J .
The endpoint a is said to be regular if a > −∞ and each of the functions