1-D Schr\"odinger operator on a star graph with nondefinite weight function

TL;DR

This paper investigates the spectral properties of an indefinite Schrödinger operator on a star graph, demonstrating its similarity to a selfadjoint operator and characterizing its eigenfunctions and spectrum.

Contribution

It provides a novel analysis of the indefinite Kirchhoff Laplacian on a star graph, showing its similarity to a selfadjoint operator and describing its eigenfunctions and spectrum.

Findings

Operator is similar to a selfadjoint operator in Hilbert space

Eigenfunctions form a Riesz basis

Complete description of the point spectrum

Abstract

On a star graph $G$ with $n = n_+ + n_-$ edges of unit length, we study the operator $-\frac{\mathrm{d}^2}{\mathrm{d} x^2}$ on $n_+$ and $\frac{\mathrm{d}^2}{\mathrm{d} x^2}$ on $n_-$ edges equipped with Dirichlet boundary conditions at the outer vertices and a Kirchhoff condition at the central vertex. We study the spectral properties of the corresponding indefinite Kirchhoff Laplacian on $G$ and we show that it is similar to a selfadjoint operator in the Hilbert space $L^2(G)$ and that its eigenfunctions form a Riesz basis. Furthermore, we give a complete description of the point spectrum.