Singular perturbations of Laplace operator and their resolvents

TL;DR

This paper investigates the spectral properties of Laplace operators on manifolds with singular perturbations, deriving resolvent formulas and analyzing the spectrum of associated boundary value problems.

Contribution

It introduces a framework for analyzing Laplace operators with singular perturbations and derives Krein's resolvent formula for these operators.

Findings

Krein's formula for resolvent differences is established.

Spectral properties of the non-smooth Bitsadze-Samarski problem are characterized.

Invertible restrictions of the Laplace operator are well-defined on the perturbed manifold.

Abstract

An inner closed (without boundary) smooth manifold of a lower dimension is cut from a multidimensional ball. In this region, invertible restrictions of the Laplace operator are well defined. In particular, the well-posed non-smooth Bitsadze-Samarski\u{i} problem for the Laplace equation is defined. Moreover, we obtain M.G. Krein's formula for the trace of the difference of resolvents of the studied operators. We prove assertions on the spectrum of the non-smooth Bitsadze-Samarski\u{i} problem.