  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.
