Symmetrized Perturbation Determinants and Applications to Boundary Data Maps and Krein-Type Resolvent Formulas

TL;DR

This paper develops an abstract framework for symmetrized Fredholm perturbation determinants, linking them to boundary data maps for Schrödinger operators, and derives trace formulas for resolvent differences based on boundary conditions.

Contribution

It introduces a novel reduction of symmetrized perturbation determinants to boundary data maps and establishes trace formulas for Schrödinger operators with different boundary conditions.

Findings

Reduction of perturbation determinants to boundary data maps

Derivation of trace formulas for resolvent differences

Extension of classical trace formulas to boundary condition variations

Abstract

The aim of this paper is twofold: On one hand we discuss an abstract approach to symmetrized Fredholm perturbation determinants and an associated trace formula for a pair of operators of positive-type, extending a classical trace formula. On the other hand, we continue a recent systematic study of boundary data maps, that is, 2 \times 2 matrix-valued Dirichlet-to-Neumann and more generally, Robin-to-Robin maps, associated with one-dimensional Schr\"odinger operators on a compact interval [0,R] with separated boundary conditions at 0 and R. One of the principal new results in this paper reduces an appropriately symmetrized (Fredholm) perturbation determinant to the 2\times 2 determinant of the underlying boundary data map. In addition, as a concrete application of the abstract approach in the first part of this paper, we establish the trace formula for resolvent differences of self-adjoint Schr\"odinger operators corresponding to different (separated) boundary conditions in terms of boundary data maps.