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

Abstract

We study asymptotic distribution of eigen-values $\omega$ of a quadratic operator polynomial of the following form $({\omega }^2-L(\omega )){\varphi }_{\omega }=0$, 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.

latex, 20 pages

References in corpus (3)

• Quantum fluctuations in brane-world inflation without inflaton on the brane
• Statistical Mechanics, Gravity, and Euclidean Theory
• Thermodynamics, Euclidean Gravity and Kaluza-Klein Reduction

Cited by in corpus (6)

• Heat kernel expansion: user's manual
• Quantum corrections to the noncommutative kink
• A Gutzwiller trace formula for stationary space-times
• On space-time noncommutative theories at finite temperature
• Divergences in the vacuum energy for frequency-dependent interactions
• Quantum aspects of a noncommutative supersymmetric kink