Nonself-adjoint Three-Dimensional Operators with Point Interactions Supported by Curves

Abstract

In a general three-dimensional domain we consider a second order differential operator with variable coefficients subject to an arbitrary boundary condition; the operator is supposed to be \(m\)-sectorial. We show how to add a point interaction supported by a given curve \( \Gamma \) with a complex-valued coupling function to this operator. We provide a rigorous way for defining such an operator with point interaction and study its basic properties. Our approach is based on the main idea used to define the Laplacian in the three-dimensional space on a given curve, but we modify this idea quite essentially avoiding at the same time any implicit Green functions. Our definition provides an effective tool for studying general operators in three-dimensional domains with point interactions on given curves.