A Jost-Pais-type reduction of Fredholm determinants and some applications

Abstract

We study the analog of semi-separable integral kernels in $\cH$ of the type {equation*} K(x,x')={cases} F_1(x)G_1(x'), & a<x'< x< b, \\ F_2(x)G_2(x'), & a<x<x'<b, {cases} {equation*} where $-\infty \le a<b\le \infty$, and for a.e.\ $x\in (a,b)$, $F_j (x) \in \cB_2(\cH_j,\cH)$ and $G_j(x) \in \cB_2(\cH,\cH_j)$ such that $F_j(\cdot )$ and $G_j(\cdot )$ are uniformly measurable, and {equation*} \|F_j(\cdot)\|_{\cB_2(\cH_j,\cH)} \in L^2((a,b)), \; \|G_j (\cdot)\|_{\cB_2(\cH,\cH_j)} \in L^2((a,b)), \quad j=1,2, {equation*} with $\cH$ and $\cH_j$, $j=1,2$, complex, separable Hilbert spaces. Assuming that $K(\cdot ,\cdot )$ generates a trace class operator $\bsK$ in $L^2((a,b);\cH)$, we derive the analog of the Jost-Pais reduction theory that succeeds in proving that the Fredholm determinant ${\det}_{L^2((a,b);\cH)}(\bsI - α\bsK)$, $\alpha \in \mathbb{C}$, naturally reduces to appropriate Fredholm determinants in the Hilbert spaces $\cH$ (and $\cH_1 \oplus \cH_2$). Explicit applications of this reduction theory are made to Schrödinger operators with suitable bounded operator-valued potentials. In addition, we provide an alternative approach to a fundamental trace formula first established by Pushnitski which leads to a Fredholm index computation of a certain model operator.

50 pages; some typos removed