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

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 $\{\lambda_k(0)\}$.