On Schr\"odinger operators with $\delta'$-potentials supported on star graphs

TL;DR

This paper investigates the spectral characteristics of 2D Schr"odinger operators with -potentials on star graphs, detailing the essential spectrum and conditions for finite discrete spectrum, including eigenvalue asymptotics for two-branch stars.

Contribution

It provides a comprehensive analysis of the spectral properties of Schr"odinger operators with -potentials on star graphs, including new asymptotic results for eigenvalues.

Findings

Characterization of the essential spectrum.

Conditions for non-trivial finite discrete spectrum.

Asymptotic behavior of eigenvalues for two-branch star graphs.

Abstract

The spectral properties of two-dimensional Schr\"odinger operators with $\delta'$-potentials supported on star graphs are discussed. We describe the essential spectrum and give a complete description of situations in which the discrete spectrum is non-trivial but finite. A more detailed study is presented for the case of a star graph with two branches, in particular, the small angle asymptotics for the eigenvalues is obtained.