Correctness of the definition of the Laplace operator with delta-like potentials

TL;DR

This paper provides a rigorous definition of the Laplace operator with delta-like potentials, explores its resolvent properties, and derives formulas including the Krein formula for these resolvents.

Contribution

It introduces a correct mathematical framework for the Laplace operator with delta-like potentials and derives explicit resolvent formulas, including Krein's formula.

Findings

Established a proper definition of the operator with delta-like potentials.

Derived explicit formulas for the resolvent of the operator.

Obtained Krein's formula for the resolvent in this context.

Abstract

In this paper, we give a correct definition of the Laplace operator with delta-like potentials. Correctly solvable pointwise perturbation is investigated and formulas of resolvent are described. We study some properties of the resolvent. In particular, we obtain Krein formula for these resolvents.