An abstract framework for interior-boundary conditions

TL;DR

This paper develops an abstract framework for interior-boundary conditions, focusing on self-adjoint operators, and provides criteria, formulas, and classifications relevant to quantum field theory applications.

Contribution

It introduces a general abstract setting for interior-boundary conditions, including self-adjointness criteria and resolvent formulas, advancing the mathematical understanding of boundary conditions in quantum physics.

Findings

Established self-adjointness criteria for operators with interior-boundary conditions

Derived resolvent formulas for these operators

Provided a classification theorem for interior-boundary conditions

Abstract

In a configuration space whose boundary can be identified with a subset of its interior, a boundary condition can relate the behaviour of a function on the boundary and in the interior. Additionally, boundary values can appear as additive perturbations. Such boundary conditions have recently provided insight into problems form quantum field theory. We discuss interior-boundary conditions in an abstract setting, with a focus on self-adjoint operators, proving self-adjointness criteria, resolvent formulas, and a classification theorem.