Spectral analysis of the Sturm-Liouville operator given on a system of segments

TL;DR

This paper investigates the spectral properties of a Sturm-Liouville operator on a system of segments, analyzing how small perturbations in the potential functions affect the simplicity and location of eigenvalues.

Contribution

It extends spectral analysis to Sturm-Liouville operators on a segmented domain with specific boundary conditions, showing eigenvalue stability under small potential perturbations.

Findings

Eigenvalues of the unperturbed operator are simple.

Eigenvalues of the perturbed operator remain simple.

Eigenvalues stay close to original eigenvalues under small perturbations.

Abstract

The spectral analysis of the Sturm-Liouville operator defined on a finite segment is the subject of an extensive literature. Sturm-Liouville operators on a finite segment are well studied and have numerous applications. The study of such operators already given on the system segments (graphs) was received in the works. This work is devoted to the study of operators $$(L_qy)(x)=col[-y_1''(x)+q_1(x)y_1(x), \ -y_2''(x)+q_2(x)y_2(x)],$$ where $y(x)=col[y_1(x),\ y_2(x)]\epsilon L^2(-a,0)\oplus L^2(0,b)=H, \ q_1(x), q_2(x) -$ real function $q_1\epsilon L^2(-a,0), q_2\epsilon L^2(0,b).$ Domain of definition $L_q$ has the form $$\vartheta (L_q)={y=(y_1,y_2)\epsilon H; \ y_1\epsilon W_1^2(-a,0), \ y_2\epsilon W_2^2(0,b), \ y_1'(-a)=0, \ y_2'(b)=0; \ y_2(0)+py_1'(0)=0 \ y_1(0)+py_2'(0)=0}$$ $(p\epsilon \mathbb{R}, \ p\neq 0).$ Such an operator is self-adjoint in $H.$ The work uses the methods described in work. The main result is as follows: if the $q_1, q_2$ are small (the degree of their smallness is determined by the parameters of the boundary conditions and the numbers $a, b$), then the eigenvalues $\{\lambda_k(0)\}$ of the unperturbed operator $L_0$ are simple, and the eigenvalues $\{\lambda_k(q)\}$ of the perturbed operator $L_q$ are also simple and located small in the vicinity of the points $\{\lambda_k(0)\}$.