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

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), F_j (x) ∈ B₂ (H_j, H) and G_j(x) ∈ B₂(H,H_j) such that F _j(·) and G _j(·) are uniformly measurable, and (Formula Presented) with H and H_j, 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 det_L2((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.