2007/05/24 by Fritz Gesztesy, Marius Mitrea, Gesztesy, Fritz +3
Computer Science · Mathematics · Physics and Astronomy · #34L40. #47G10 #Advanced Mathematical Modeling in Engineering #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Numerical methods in inverse problems #Primary: 47B10 #Secondary: 34B27 #Spectral Theory (math.SP) #Spectral Theory in Mathematical Physics #math-ph #math.MP #math.SP #msc:34B27 #msc:34L40. #msc:47B10 #msc:47G10
paper · pdf · doi:10.48550/arxiv.0705.3510
40 pages. To appear in J. Funct. Anal
arxiv created 2007/05/24 · openalex publication_date 2007/05/24 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We explore the extent to which a variant of a celebrated formula due to Jost and Pais, which reduces the Fredholm perturbation determinant associated with the Schrödinger operator on a half-line to a simple Wronski determinant of appropriate distributional solutions of the underlying Schrödinger equation, generalizes to higher dimensions. In this multi-dimensional extension the half-line is replaced by an open set Ω⊂\bbRn, n∈\bbN, n≥ 2, where Ω has a compact, nonempty boundary ∂Ω satisfying certain regularity conditions. Our variant involves ratios of perturbation determinants corresponding to Dirichlet and Neumann boundary conditions on ∂Ω and invokes the corresponding Dirichlet-to-Neumann map. As a result, we succeed in reducing a certain ratio of modified Fredholm perturbation determinants associated with operators in L2(Ω; dn x), n∈\bbN, to modified Fredholm determinants associated with operators in L2(∂Ω; dn-1σ), n≥ 2. Applications involving the Birman-Schwinger principle and eigenvalue counting functions are discussed.