Jiong Sun

The finite spectrum of Sturm–Liouville problems with transmission conditions

For any positive integer n and a set of positive integers \(m_{i}\), \(i=1,2,\ldots ,n+1\), we construct a class of regular Sturm–Liouville problems with n transmission conditions, which have exactly \(\sum ^{n+1}_{i=1}m_{i}+n+1\) eigenvalues. And further we show that these \(\sum ^{n+1}_{i=1}m_{i}+n+1\) eigenvalues can be distributed arbitrarily throughout the complex plane in the non-self-adjoint case and anywhere along the real line in the self-adjoint case.

Matrix representations of Sturm-Liouville problems with transmission conditions

We identify a class of Sturm-Liouville equations with transmission conditions such that any Sturm-Liouville problem consisting of such an equation with transmission condition and an arbitrary separated or real coupled self-adjoint boundary condition has a representation as an equivalent finite dimensional matrix eigenvalue problem. Conversely, given any matrix eigenvalue problem of certain type and an arbitrary separated or real coupled self-adjoint boundary condition and transmission condition, we construct a class of Sturm-Liouville problems with this specified boundary condition and transmission condition, each of which is equivalent to the given matrix eigenvalue problem.

Matrix representations of Sturm–Liouville problems with coupled eigenparameter-dependent boundary conditions

Abstract We study the matrix representations of Sturm–Liouville problems with coupled eigenparameter-dependent boundary conditions with a finite spectrum. We prove for any positive integer n, the considered problems have at most n + 3 eigenvalues, and show that this kind of Sturm–Liouville problems with coupled eigenparameter-dependent boundary conditions is equivalent to a class of matrix eigenvalue problems in the sense that they have exactly the same eigenvalues.

Survey Article: Self-adjoint ordinary differential operators and their spectrum

We survey the theory of ordinary selfadjoint differential operators in Hilbert space and their spectrum. Such an operator is generated by a symmetric differential expression and a boundary condition. We discuss the very general modern theory of these symmetric expressions which enlarges the class of these expressions by many dimensions and eliminates the smoothness assumptions required in the classical case as given, e.g., in the celebrated books by Coddington and Levinson and Dunford and Schwartz. The boundary conditions are characterized in terms of squareintegrable solutions for a real value of the spectral parameter, and this characterization is used to obtain information about the spectrum. Many of these characterizations are quite recent and widely scattered in the literature, some are new. A comprehensive review of the deficiency index (which determines the number of independent boundary conditions required in the singular case) is also given for an expression M and for its powers. Using the modern theory mentioned above, these powers can be constructed without any smoothness conditions on the coefficients.

Fourth order boundary value problems with finite spectrum

Abstract For every positive integer m we construct a class of regular self-adjoint and non-self-adjoint fourth order boundary value problems with at most 3 m + 1 eigenvalues, counting multiplicity.

Finite spectrum of 2nth order boundary value problems

Abstract For any even positive integer 2 n and any positive integer m we construct a class of regular self-adjoint and non-self-adjoint boundary value problems whose spectrum consists of at most ( 2 n − 1 ) m + 1 eigenvalues. Our main result reduces to previously known results for the cases n = 1 and n = 2 . In the self-adjoint case with separated boundary conditions this upper bound can be improved to n ( m + 1 ) .

THE DISCRETENESS OF SPECTRUM FOR HIGHER-ORDER DIFFERENTIAL OPERATORS IN WEIGHTED FUNCTION SPACES

In this paper we consider the discreteness of spectrum for higher-order differential operators in the weighted function spaces. Using the method of the embedding theorems of weighted Sobolev spaces

Characterization of domains of self-adjoint ordinary differential operators

H_p^n

The classification of self-adjoint boundary conditions: Separated, coupled, and mixed

in weighted spaces

Two-interval Sturm–Liouville operators in modified Hilbert spaces

L_{s,r}