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

Abstract

We study asymptotic distribution of eigen-values ω of a quadratic operator polynomial of the following form (ω 2 − L(ω))φ ω = 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 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 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.

Quantum corrections to the noncommutative kink

“…Since the potential in (9) depends on the frequency ω we are dealing with a non-linear spectral problem. To analyze the spectral density we use a method developed initially in [20][21][22] and then adapted to NC theories in [18]. Let us consider an auxiliary eigenvalue problem…”

“…The spectral density ρ(σ, ω) taken at coinciding arguments is not the density ρ(ω) of our initial spectral problem (9). As demonstrated in [20][21][22] (see also [18] for a discussion in the framework of NC theories) one has to construct another density ̺(σ, ω) which is related to a heat-kernel like object…”

Divergences in the vacuum energy for frequency-dependent interactions

“…By using Eqs. (11), (12) and (21), one obtains the part of the spectral density for the nonlinear spectral problem, which is generated by the tachionic part of the auxiliary problem:…”

“…By using Eqs. (12) and (21) one obtains an expression for the spectral density of the non-linear spectral problem…”

“…if we are dealing with a non-linear spectral problem. If the dependence of D on ω is rather mild, as happens on stationary but non-static backgrounds [10,11,12], the divergences can still be expressed through the asymptotics of the heat kernel for D where the dependence on ω is frozen. In time-space noncommutative theories, which also lead to non-linear spectral problems, the divergences of the vacuum energy are also expressible through the polynomials of the background field, but the structure of these polynomials becomes more complicated [13,14].…”