Relative trace formulas for obstacle scattering with Neumann and transmission boundary conditions

TL;DR

This paper develops a relative trace formula for obstacle scattering with Neumann and transmission boundary conditions, linking it to Casimir energy and extending mathematical understanding of the Casimir effect.

Contribution

It introduces a new relative trace formula for Neumann and transmission boundary conditions, generalizing previous results for Dirichlet conditions and connecting to physical Casimir energy.

Findings

Trace formula for Neumann and transmission conditions established

Connection between trace formula and Casimir energy demonstrated

Rigorous Lifshitz formula recovered in one-dimensional case

Abstract

We consider the case of scattering by several obstacles in $\mathbb{R}^d$ for $d \geq 2$. We establish a relative trace formula for Neumann and transmission boundary conditions analogous to the one obtained in arXiv:2002.07291 for Dirichlet boundary conditions. In the case of $f(x) = x^{1/2}$ the trace has the interpretation of the Casimir energy of the obstacle configuration. In the one-dimensional case, we recover a rigorous version of the Lifshitz formula for the Casimir energy of parallel plates with frequency-independent electric permittivity and magnetic permeability. We thereby strengthen the mathematical foundations of the Casimir effect and demonstrate the flexibility of the rigorous approach established in arXiv:2104.09763 and arXiv:2002.07291.