Interior-Boundary Conditions for Schrodinger Operators on Codimension-1 Boundaries

TL;DR

This paper develops a general form of interior-boundary conditions for Schrödinger operators on codimension-1 boundaries, facilitating quantum field theory Hamiltonians without renormalization.

Contribution

It introduces a conjectured general form of IBCs for Schrödinger operators on codimension-1 boundaries, simplifying the analysis of particle interactions.

Findings

Proposes a conjectured general form of IBCs for Schrödinger operators.

Focuses on boundaries with codimension 1 for simplicity.

Lays groundwork for future extensions to higher codimension boundaries.

Abstract

Interior-boundary conditions (IBCs) are boundary conditions on wave functions for Schr\"odinger equations that allow that probability can flow into (and thus be lost at) a boundary of configuration space while getting added in another part of configuration space. IBCs are of particular interest because they allow defining Hamiltonians involving particle creation and annihilation (as used in quantum field theories) without the need for renormalization or ultraviolet cut-off. For those Hamiltonians, the relevant boundary has codimension 3. In this paper, we develop (what we conjecture is) the general form of IBCs for the Laplacian operator (or Schr\"odinger operators), but we focus on the simpler case of boundaries with codimension 1.