Symmetrized perturbation determinants and applications to boundary data maps and Krein-type resolvent formulas

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.

Effective computation of traces, determinants, and ζ-functions for Sturm–Liouville operators

“…Here any choice of branch cut of the normal operator (B − zI H ) 1/2 (employing the spectral theorem) is permissible. The first equality in (2.15) follows as in the proof of [23,Theorem 2.8]. (The details are actually a bit simpler now since A, B are selfadjoint and bounded from below, and hence of positive type after some translation, which is the case considered in [23]).…”

“…The first equality in (2.15) follows as in the proof of [23,Theorem 2.8]. (The details are actually a bit simpler now since A, B are selfadjoint and bounded from below, and hence of positive type after some translation, which is the case considered in [23]). For completeness we mention that the second equality in (2.15) can be arrived at as follows: Employing the commutation formula (cf., [9]),…”

“…Given Hypothesis 2.4 it has been proven in [23] (actually, in a more general context) that for z ∈ C\[inf(σ(A) ∪ σ(B)), ∞),…”

Trace formulas for additive and non-additive perturbations

“…Note that a different approach to perturbation determinants for singular perturbations was proposed in [26,31]. It is based on the use of positive-type operators and its applicability requires that either the square root domain of H is contained in that one of H or vice versa.…”

On index theory for non‐Fredholm operators: A (1 + 1)‐dimensional example

“…Moreover, assume that for some (and hence for all) z 0 ∈ C\R, We note that these considerations naturally extend to more complex situations where A = A 0 + q B, A n = A 0,n + q B n , n ∈ N, are defined as quadratic form sums of A 0 and B and A 0,n and B n (without assuming any correlation between the domains of A, A n and A 0 ), and the modified Fredholm determinants are replaced by symmetrized ones as in Theorem A.1, see, for instance, [13], [14], and [17]. Since we do not need this at this point, we omit further details.…”