Heat-kernel and resolvent asymptotics for Schroedinger operators on metric graphs

TL;DR

This paper analyzes the spectral properties of Schroedinger operators on metric graphs, providing asymptotic expansions of the heat-semigroup trace and explicit leading coefficients, advancing understanding of quantum graph operators.

Contribution

It introduces trace-class regularizations and derives a complete asymptotic expansion of the heat-semigroup trace for Schroedinger operators on metric graphs, including explicit leading coefficients.

Findings

Complete asymptotic expansion of the heat-semigroup trace

Explicit formulas for leading coefficients

Representation of resolvents as integral operators

Abstract

We consider Schroedinger operators on compact and non-compact (finite) metric graphs. For such operators we analyse their spectra, prove that their resolvents can be represented as integral operators and introduce trace-class regularisations of the resolvents. Our main result is a complete asymptotic expansion of the trace of the (regularised) heat-semigroup generated by the Schroedinger operator. We also determine the leading coefficients in the expansion explicitly.