Resolvent expansions on hybrid manifolds

TL;DR

This paper investigates the spectral properties of Laplace-type operators on hybrid manifolds composed of 2D manifolds and 1D segments, deriving resolvent expansions and exploring inverse spectral questions.

Contribution

It introduces a method to analyze resolvent expansions for Laplace operators on hybrid manifolds and addresses inverse spectral problems in this context.

Findings

Derived large spectral parameter expansion of the trace of the second resolvent power

Provided insights into inverse spectral theory for hybrid manifolds

Established a framework for spectral analysis on complex geometric configurations

Abstract

We study Laplace-type operators on hybrid manifolds, i.e. on configurations consisting of closed two-dimensional manifolds and one-dimensional segments. Such an operator can be constructed by using the Laplace-Beltrami operators on each component with some boundary conditions at the points of gluing. The large spectral parameter expansion of the trace of the second power of the resolvent is obtained. Some questions of the inverse spectral theory are adressed.