On the extension of Fredholm determinants to the mixed multidimensional integral operators with regulated kernels

TL;DR

This paper extends classical Fredholm determinants to a broad class of multidimensional integral operators with regulated kernels, providing explicit formulas for inverses and spectral analysis relevant in wave physics.

Contribution

It introduces a maximal Banach algebra of integral operators with explicit trace and determinant constructions, generalizing classical results to multidimensional and mixed integral operators.

Findings

Constructed explicit inverse operators (resolvents).

Described the spectrum via zeroes of determinants.

Established multiple determinants corresponding to different spectral components.

Abstract

We extend the classical trace (and determinant) known for the integral operators $$ ({\mathcal I}+)\int_{[0,1)^N}{\bf A}({\bf k},{\bf x}){\bf u}({\bf x})d{\bf x} $$ with matrix-valued kernels ${\bf A}$ to the operators of the form $$ \sum_{\alpha}\int_{[0,1)^{|\alpha|}}{\bf A}({\bf k},{\bf x}_{\alpha}){\bf u}({\bf k}_{\overline{\alpha}},{\bf x}_{\alpha})d{\bf x}_{\alpha}, $$ where $\alpha$ are arbitrary subsets of the set $\{1,...,N\}$. Such operators form a Banach algebra containing simultaneously all integral operators of the dimensions $\leqslant N$. In this sense, it is a largest algebra where explicit traces and determinants are constructed. Such operators arise naturally in the mechanics and physics of waves propagating through periodic structures with various defects. We give an explicit representation of the inverse operators (resolvent) and describe the spectrum by using zeroes of the determinants. Due to the structure of the operators, we have $2^N$ different determinants, each of them describes the spectral component of the corresponding dimension.