Abstract
The aim of this paper is two-fold: 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, 2×2 matrix-valued Dirichlet-to-Neumann and more generally, Robin-to-Robin maps, associated with one-dimensional Schrödinger 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 2×2 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 Schrödinger operators corresponding to different (separated) boundary conditions in terms of boundary data maps.
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