On a class of integral systems

TL;DR

This paper investigates spectral properties of a class of two-dimensional integral systems, generalizing Krein strings, introducing Titchmarsh-Weyl coefficients, and establishing relations between dual systems' spectral data.

Contribution

It introduces a framework for analyzing spectral problems of generalized integral systems, including boundary conditions, Weyl functions, and dual system relations.

Findings

Limit point condition characterized via linear relations.

Boundary triples and Weyl functions constructed for spectral analysis.

Dual systems' Titchmarsh-Weyl coefficients related by a specific reciprocal formula.

Abstract

We study spectral problems for two--dimensional integral system with two given non-decreasing functions $R_1$, $R_2$ on an interval $[0,b)$ which is a generalization of the Krein string. Associated to this system are the maximal linear relation $T_{\max}$ and the minimal linear relation $T_{\min}$ in the space $L^2(R_2)$ which are connected by $T_{\max}=T_{\min}^*$. It is shown that the limit point condition at $b$ for this system is equivalent to the strong limit point condition for the linear relation $T_{\max}$. In the limit circle case the strong limit point condition fails to hold on $T_{\max}$ but it is still satisfied on a subspace $T_N^*$ of $T_{\max}$ characterized by the Neumann boundary condition at $b$. The notion of the principal Titchmarsh-Weyl coefficient of this integral system is introduced both in the limit point case and in the limit circle case. Boundary triples for the linear relation $T_{\max}$ in the limit point case (and for $T_{N}^*$ in the limit circle case) are constructed and it is shown that the corresponding Weyl function coincides with the principal Titchmarsh-Weyl coefficient of the integral system. The notion of the dual integral system is introduced by reversing the order of $R_1$ and $R_2$. It is shown that the principal Titchmarsh-Weyl coefficients $q$ and $\widehat q$ of the direct and the dual integral systems are related by the equality $\lambda \widehat q(\lambda) = -1/q(\lambda)$ both in the regular and the singular case.