Partial Non-stationary Perturbation Determinants for a Class of J-symmetric Operators

Abstract

We consider the partial non-stationary perturbation determinant

$$ \Delta _{H/A}^{(1)} (t): = \det \left( {e^{itA} P_1 e^{ - itH} |_{\mathcal{H}_1 } } \right),t \in \mathbb{R}.$$

Here H is a self-adjoint operator in some Krein space $\mathcal{K}$ and A is a self-adjoint operator in the Hilbert space $\mathcal{H}_1$, which is the positive component of a fundamental decomposition of $\mathcal{H}_1$ with corresponding orthogonal projection P₁. The asymptotic behavior of δ ⁽¹⁾ _H/A (t) for t → ∞ and the spectral shift function for H and its diagonal part are studied. Analogous results for the case if the underlying space is a Hilbert space were obtained in [1].