Lower bound on the spectrum of the Schr\"odinger operator in the plane with delta-potential supported by a curve

TL;DR

This paper derives explicit lower bounds for the spectrum of a Schrödinger operator with delta-potential supported on curves in the plane, considering various geometric configurations and applications to quantum graphs.

Contribution

It provides new spectral estimates for Schrödinger operators with delta-potentials on curves, including finite, infinite, and piecewise segments, with applications to cusps and leaky quantum graphs.

Findings

Explicit lower bounds for infinite and finite curves

Spectral estimates for piecewise curves based on segment parameters

Application to quantum graphs with cusps and leaks

Abstract

We consider the Schr\"odinger operator in the plane with delta-potential supported by a curve. For the cases of an infinite curve and a finite loop we give estimates on the lower bound of the spectrum expressed explicitly through the strength of the interaction and a parameter which characterizes geometry of the curve. Going further we cut the curve into finite number of pieces and estimate the bottom of the spectrum using the parameters for the pieces. As an application of the elaborated theory we consider a curve with a finite number of cusps and "leaky" quantum graphs.