Spectral Asymptotics of Eigen-value Problems with Non-linear Dependence on the Spectral Parameter

TL;DR

This paper investigates the asymptotic distribution of eigenvalues for a quadratic operator polynomial with non-linear spectral dependence, deriving spectral density asymptotics and linking them to heat kernel expansions, with applications in physics.

Contribution

It introduces a method to compute spectral asymptotics for quadratic operator polynomials with non-linear spectral dependence, connecting it to heat kernel coefficients.

Findings

Derived asymptotics of spectral density for the operator polynomial.

Showed the leading term matches that of a Laplacian in a cavity.

Provided examples in gravitational and gauge field backgrounds.

Abstract

We study asymptotic distribution of eigen-values $\omega$ of a quadratic operator polynomial of the following form $(\omega^2-L(\omega))\phi_\omega=0$, where $L(\omega)$ is a second order differential positive elliptic operator with quadratic dependence on the spectral parameter $\omega$. We derive asymptotics of the spectral density in this problem and show how to compute coefficients of its asymptotic expansion from coefficients of the asymptotic expansion of the trace of the heat kernel of $L(\omega)$. The leading term in the spectral asymptotics is the same as for a Laplacian in a cavity. The results have a number of physical applications. We illustrate them by examples of field equations in external stationary gravitational and gauge backgrounds.