Fredholm determinants, Evans functions and Maslov indices for partial differential equations

TL;DR

This paper develops a multidimensional analogue of the Evans function using Fredholm determinants, enabling spectral analysis of non-self-adjoint elliptic operators and connecting it to the Maslov index for deeper insights.

Contribution

It introduces a novel multidimensional Evans function via Fredholm determinants, extending spectral tools to non-self-adjoint elliptic operators and linking to the Maslov index.

Findings

Constructed a multidimensional Evans function as a Fredholm determinant.

Established relations between eigenvalue multiplicities and operator pencils.

Connected the construction to the Maslov index, providing new spectral insights.

Abstract

The Evans function is a well known tool for locating spectra of differential operators in one spatial dimension. In this paper we construct a multidimensional analogue as the modified Fredholm determinant of a ratio of Dirichlet-to-Robin operators on the boundary. This gives a tool for studying the eigenvalue counting functions of second-order elliptic operators that need not be self-adjoint. To do this we use local representation theory for meromorphic operator-valued pencils, and relate the algebraic multiplicities of eigenvalues of elliptic operators to those of the Robin-to-Robin and Robin-to-Dirichlet operator pencils. In the self-adjoint case we relate our construction to the Maslov index, another well known tool in the spectral theory of differential operators. This gives new insight into the Maslov index and allows us to obtain crucial monotonicity results by complex analytic methods.