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

• Published in: Integral equations and operator theory 2014-07, Vol.79 (3), p.389-447
• Main Authors: Carey, Alan, Gesztesy, Fritz, Potapov, Denis, Sukochev, Fedor, Tomilov, Yuri
• Format: Article
• Language: English
• Subjects: Analysis; Mathematics; Mathematics and Statistics
• ISSN: 0378-620X; 1420-8989
• DOI: 10.1007/s00020-014-2150-0

We study the analog of semi-separable integral kernels in H of the type K ( x , x ′ ) = F 1 ( x ) G 1 ( x ′ ) , a < x ′ < x < b , F 2 ( x ) G 2 ( x ′ ) , a < x < x ′ < b , where - ∞ ⩽ a < b ⩽ ∞ , and for a.e. x ∈ ( a , b ) , F j ( x ) ∈ B 2 ( H j , H ) and G j ( x ) ∈ B 2 ( H , H j ) such that F j (·) and G j (·) are uniformly measurable, and | | F j ( · ) | | B 2 ( H j , H ) ∈ L 2 ( ( a , b ) ) , | | G j ( · ) | | B 2 ( H , H j ) ∈ L 2 ( ( a , b ) ) , j = 1 , 2 , with H and H j , j  = 1, 2, complex, separable Hilbert spaces. Assuming that K (·, ·) generates a trace class operator K in L 2 ( ( a , b ) ; H ) , we derive the analog of the Jost–Pais reduction theory that succeeds in proving that the Fredholm determinant det L 2 ( ( a , b ) ; H ) ( I − α K ), α ∈ C , naturally reduces to appropriate Fredholm determinants in the Hilbert spaces H (and H 1 ⊕ H 2 ). 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.