Schrödinger operators with singular interactions: a model of tunnelling resonances

Abstract

We discuss a generalized Schrödinger operator in We analyze the discrete spectrum, and furthermore, we show that the resonance problem in this setting can be explicitly solved; by BirmanSchwinger method it is cast into a form similar to the Friedrichs model.

Non-relativistic and relativistic scattering by short-range potentials

“…However, the solution obtained by regularization is not unique and has been studied in more detail by Zolotaryuk et al [3] and Zolotaryuk [4]. Scattering potentials with a higher degree of singularity than the d potential were considered by Exner & Kondej [5], Zolotaryuk et al [6] and Zolotaryuk [7].…”

On Relations Between Stable and Zeno Dynamics in a Leaky Graph Decay Model

“…By assumption there are no embedded eigenvalues (cf. Remark 4.1) and by [9] also the singularly continuous component is void, hence the second term is associated solely with σ ac (H α,β ). Let us first look at this contribution to the reduced evolution.…”

“…It is now a standard thing to check thatḢ α,β is essentially self-adjoint [9]; we identify its closure denoted as H α,β with the formal Hamiltonian (3.1). To find the resolvent of H α,β we start from R(z) = (−∆−z) −1 which is for any z ∈ C \ [0, ∞) an integral operator with the kernel…”

“…[9]. Using this notation we can rewrite the matrix elements of (Γ 10 Γ −1 00 Γ 01 ) (·) (·) in the following form,…”

“…As we have said the absence of the straight-line interaction can be regarded in a sense as putting the line to an infinite distance from the points, thus corresponding tob = 0. In this case we have Now one proceeds as in [9] checking that the hypotheses of the implicit-function theorem are satisfied; then the equation η(b, z) = 0 has for all the b i small enough just m zeros which admit the following weak-coupling asymptotic expansion,…”