Spectral asymptotics in eigenvalue problems with nonlinear dependence on the spectral parameter

We study the asymptotic distribution of eigenvalues ω of a quadratic operator polynomial of the form (ω2 - L(ω))φω = 0, where L(ω) is a second-order differential positive elliptic operator with quadratic dependence on the spectral parameter ω. We derive asymptotics of the spectral density in this problem and show how to compute coefficients of its asymptotic expansion from the coefficients of the asymptotic expansion of the trace of the heat kernel of L(ω). The leading term in the spectral asymptotics is the same as that 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