Schrödinger Operators with δ and δ′-Potentials Supported on Hypersurfaces

Abstract

Abstract. Self-adjoint Schrödinger operators with δ and δ ′ -potentials supported on a smooth compact hypersurface are defined explicitly via boundary conditions. The spectral properties of these operators are investigated, regularity results on the functions in their domains are obtained, and analogues of the Birman-Schwinger principle and a variant of Krein's formula are shown. Furthermore, Schatten-von Neumann type estimates for the differences of the powers of the resolvents of the Schrödinger operators with δ and δ ′ -potentials, and the Schrödinger operator without a singular interaction are proved. An immediate consequence of these estimates is the existence and completeness of the wave operators of the corresponding scattering systems, as well as the unitary equivalence of the absolutely continuous parts of the singularly perturbed and unperturbed Schrödinger operators. In the proofs of our main theorems we make use of abstract methods from extension theory of symmetric operators, some algebraic considerations and results on elliptic regularity.

Boundary triples for Schrödinger operators with singular interactions on hypersurfaces

“…Moreover, the spectral properties of A δ,α can be described with the help of the perturbation term (Θ δ,α − M (λ)) −1 . We mention that in the context of the more general notion of quasi boundary triples and their Weyl functions from [5,7] a similar approach as in this note and closely related results can be found in [8,9]; we also refer to [26,27,29,31,35,[38][39][40] for other methods in extension theory of elliptic differential operators.…”

“…We are now able to obtain some immediate and important consequences from the previous considerations, well-known results for boundary triples and Weyl functions [14,15] and the resolvent estimates in [4,8]. .…”

“…We also recall that the essential spectrum of the operator A δ,α is [0, ∞) and that the negative spectrum consists of finitely many eigenvalues of finite multiplicity; see [8,10]. In the following we fix some point η such that…”

“…Let S be the symmetric operator in (9), let {L 2 (C), Υ 0 , Υ 1 } be the boundary triple for S * in Lemma 3.1 and fix η as in (8).…”

“…where δ C denotes the δ-distribution supported on C. A precise definition of the self-adjoint operator A δ,α in terms of boundary or interface conditions is given at the beginning of Section 3 below; see also [8,10] for an equivalent definition via quadratic forms. Schrödinger operators with δ-interactions are frequently used in mathematical physics to model interactions of quantum particles; we refer to the monographs [1] and [23], to the review article [18] and to [2, 3, 8, 16, 17, 19-22, 24, 25, 32, 33, 36, 37, 42] for a small selection of related papers on spectral analysis of such operators.…”