A Jost-Pais-Type Reduction of Fredholm Determinants and Some Applications

Abstract

We study the analog of semi-separable integral kernels in H of the type (Formula Presented) where -∞ ≤ a < b ≤ ∞, and for a.e. x ∈ (a, b), Fj (x) ∈ B2 (Hj, H) and Gj(x) ∈ B2(H,Hj) such that F j(·) and G j(·) are uniformly measurable, and (Formula Presented) with H and Hj, j = 1, 2, complex, separable Hilbert spaces. Assuming that K(·, ·) generates a trace class operator K in L2((a, b);H), we derive the analog of the Jost-Pais reduction theory that succeeds in proving that the Fredholm determinant detL2((a,b);H) (I - α K), α ∈ ℂ, naturally reduces to appropriate Fredholm determinants in the Hilbert spaces H (and H1 ⊕ H2). Explicit applications of this reduction theory to Schrödinger operators with suitable bounded operator-valued potentials are made. 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. © 2014 Springer Basel.