Variations on a Theme of Jost and Pais
Fritz Gesztesy, Marius Mitrea, and Maxim Zinchenko

TL;DR
This paper extends a formula relating Fredholm perturbation determinants for Schrödinger operators from one dimension to higher dimensions, involving boundary conditions and Dirichlet-to-Neumann maps, with applications to eigenvalue analysis.
Contribution
It generalizes a known one-dimensional determinant reduction formula to higher dimensions using boundary operators and Dirichlet-to-Neumann maps.
Findings
Reduced ratios of Fredholm determinants to boundary operators.
Connected perturbation determinants with boundary conditions.
Discussed applications to eigenvalue counting and Birman-Schwinger principle.
Abstract
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 Schr\"odinger operator on a half-line to a simple Wronski determinant of appropriate distributional solutions of the underlying Schr\"odinger equation, generalizes to higher dimensions. In this multi-dimensional extension the half-line is replaced by an open set , , , where has a compact, nonempty boundary satisfying certain regularity conditions. Our variant involves ratios of perturbation determinants corresponding to Dirichlet and Neumann boundary conditions on 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…
Peer Reviews
No public reviews on file for this paper yet. If you reviewed it on a platform where reviews are public (OpenReview, ICLR, NeurIPS, ICML), you can paste yours below so the community can read it here.
Videos
No videos yet. Explain this paper in a talk, walkthrough, or lecture? Add one.
Taxonomy
TopicsSpectral Theory in Mathematical Physics · Numerical methods in inverse problems · Advanced Mathematical Modeling in Engineering
