Boundary Data Maps for Schrödinger Operators on a Compact Interval

Abstract

Abstract. We provide a 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. Most of our results are formulated in the non-self-adjoint context.Our principal results include explicit representations of these boundary data maps in terms of the resolvent of the underlying Schrödinger operator and the associated boundary trace maps, Krein-type resolvent formulas relating Schrödinger operators corresponding to different (separated) boundary conditions, and a derivation of the Herglotz property of boundary data maps (up to right multiplication by an appropriate diagonal matrix) in the special self-adjoint case.

Boundary data maps and Krein's resolvent formula for Sturm-Liouville operators on a finite interval

“…In addition, we note that (4.2) and (4.9) imply [7]). In this paper we will pursue an alternative route based on Krein's resolvent formula in Corollary 4.12.…”

“…Unfortunately, the computation of S(A ′′ , B ′′ , A ′′′ , B ′′′ ) −1 appears to be too elaborate to pursue explicit formulas for (4.42). (The special case of separated boundary conditions, however, is sufficiently simple, and in this case S(A ′′ , B ′′ , A ′′′ , B ′′′ ) −1 was explicitly computed in [7]). We now turn our attention to a derivation of a representation for the general boundary data map Λ A ′ ,B ′ A,B (z) in terms of the resolvent (H A,B − zI (a,b) ) −1 and the boundary trace map γ A ′ ,B ′ (cf.…”

“…For A, B ∈ C 2×2 satisfying (2.7), assume that U A,B : N + → N − denotes the unique linear isometric isomorphism with 7) guaranteed to exist by Theorem 6.1 (ii). Suppose that U A,B denotes the matrix representation of U A,B with respect to the bases B ± .…”

“…The principal theme developed in this paper concerns a detailed treatment of generalizations of the spectral parameter dependent Dirichlet-to-Neumann map for (three-coefficient) Sturm-Liouville operators on the finite interval (a, b) to that for all self-adjoint boundary conditions. While the earlier treatments of boundary data maps in [7] and [12] focused on the special case of separated boundary conditions at a and b, this paper now treats the case of all self-adjoint boundary conditions in a unified matter. Applications of the formalism discussed in this paper include the precise connections with Krein's resolvent formula for self-adjoint extensions of the underlying minimal Sturm-Liouville operator (parametrized in terms of boundary conditions); in particular, we will describe in detail the connections with Krein's extension of the minimal operator.…”

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

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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 in [14], 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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