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

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

2002-01-28

arxiv.org/abs/hep-th/0201219v1

Class.Quant.Grav. 19 (2002) 3635-3652; Erratum-ibid. 20 (2003) 565

Physics

High Energy Physics

High Energy Physics - Theory

latex, 20 pages

Scientific paper

10.1088/0264-9381/19/14/306

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.

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