Partial Non-stationary Perturbation Determinants

Abstract

A partial non-stationary perturbation determinant Δ₁ (t) is defined as follows:

$$\begin{array}{*{20}{c}} {{\Delta _1}(t): = \det \left( {{e^{itA}}{P_1}{e^{ - itH}}{|_{{\mathcal{H}_1}}}} \right),}&{t > 0;} \end{array}$$

here A is a self-adjoint operator in some Hilbert space ${\mathcal{H}_1}$, H is a self-adjoint operator in a larger Hilbert space $\mathcal{H} \supset {\mathcal{H}_1}$, P ₁ i the orthogonal projection in $\mathcal{H}$ onto ${\mathcal{H}_1}$ and ${P_1}(H - A){|_{{\mathcal{H}_1}}}$ is a trace class operator in ${\mathcal{H}_1}$. If the operator P ₁ H(I- P ₁) is finite-dimensional, Δ₁(t) is expressed by the resolvent kernel of a system of Fredholm integral equations on (0, t) of second kind. Moreover, in a particular situation the asymtotic behavior of Δ₁(t) for t → ∞ is studied.