Spectral analysis of integro-differential equations arising in thermal physics (Q2135112)

This paper is devoted to the spectrum of an operator function arising in the study of the following abstract second-order integro-differential equation in a separable Hilbert space \(H\): \[ \frac{d^2u(t)}{dt^2} + \int \limits ^t_0 Q(t - s)\frac{du(s)}{ds} ds + A^2u(t) - \int \limits ^t_0 K(t - s)A^2 2u(s) ds = f(t), \tag{1} \] with \(t \in \mathbb{R}_+\) and initial conditions \(u(+0) = \varphi_0\), \(u^{\prime}(+0) = \varphi_1\). Here, \(A \) is a linear operator on \( H\) and \(A: \operatorname{Dom}(A) \to H\) is a self-adjoint positive definite operator with compact inverse. The author is mainly interested in the spectral analysis of the operator function resulting from applying the Laplace transform to the left-hand side of Equation~(1). In particular, the localization of the spectrum of this operator function is obtained. The asymptotics of the non-real part of the spectrum is constructed.