Abstract
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.
Highlights
The study of spectral problems for linear ordinary differential equations 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 [6] for generalizations. This theorem gives a one-to-one correspondence between the self-adjoint differential operators in a Hilbert function space representing a given quasi differential equation (QDE) and the unitary isometries on an appropriate finite-dimensional subspace
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]
Summary
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]. 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, 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. We have the equalities among three multiplicities of an eigenvalue for a self-adjoint linear differential operator
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