Resonance asymptotics in the generalized Winter model

Abstract

We consider a modification of the Winter model describing a quantum particle in presence of a spherical barrier given by a fixed generalized point interaction. It is shown that the three classes of such interactions correspond to three different types of asymptotic behaviour of resonances of the model at high energies.Models with contact interactions are popular because they allow us to study properties of quantum systems in a framework which makes explicit solutions possible. It was found in the beginning of the eighties that the usual δ interaction on the line has a counterpart, named not quite fortunately δ ′ , and a little later the complete four-parameter class of the generalized point interactions (GPI's) was introduced [1,2]. Properties of these interactions are now well understood -see [3] for a rather complete bibliography.The GPI's fall into different classes according to their behaviour at low and high energies. The most simple manifestation can be found in scattering. While a δ-type barrier behaves as a "usual" regular potential becoming transparent at high energies, the δ ′ -like one on the contrary decouples asymptotically in the same limit. In addition, there is an intermediate class for which both the reflection and transmission amplitudes have nonzero limits as the energy tends to zero or infinity [4]. Furthermore, in Kronig-Penney-type models describing periodic arrays of such interactions, the indicated classes differ by the gap behaviour at high energies; this has important consequences for spectral nature of the corresponding Wannier-Stark systems [5,6].

Generalized interactions supported on hypersurfaces

“…Let us consider a further class of generalized interactions, denoted as generalized interactions of δ-type in the one-dimensional case in [12]. Theorem 4.4.…”

“…We show that a certain class of generalized interactions admits an estimate from below against the δ ′ -operator of an appropriate strength. This class of interactions, where β = 0 and Re γ may be nontrivial, is called the intermediate class in [12]. For a function β : Σ → R such that 1/ β is measurable and bounded we denote by −∆ δ ′ , β the selfadjoint operator in L 2 (R n ) corresponding to a δ ′ -interaction of strength β, i.e., −∆ δ ′ , β = H A with A = 0 0 0 β .…”

Solvable Models of Resonances and Decays

“…we are interested in the time evolution, ψ( r, t) = e −iHαt ψ( r, 0) for a fixed initial condition ψ( r, 0) with the support inside the ball of radius R, and the corresponding decay law P ψ (t) = R 0 |φ(r, t)| 2 dr referring to H u = L 2 (B R (0)). It is straightforward to check [AGS87] that H α has no bound states, on the other hand, it has infinitely many resonances with the widths increasing logarithmically with respect to the resonance index [EF06]; a natural idea is to employ them as a tool to expand the quantities of interest [GMM95]. In order to express reduced evolution in the way described in Sec.…”

“…). It is straightforward to check [AGS87] that H α has no bound states, on the other hand, it has infinitely many resonances with the widths increasing logarithmically with respect to the resonance index [EF06]; a natural idea is to employ them as a tool to expand the quantities of interest [GMM95]. In order to express reduced evolution in the way described in Sec.…”

Interlaced Dense Point and Absolutely Continuous Spectra for Hamiltonians With Concentric-Shell Singular Interactions

“…Recall that these types of spectral behaviour correspond to high-energy properties of a single generalized point interaction as manifested through scattering, resonances [5], etc. There is one more difference from standard Floquet theory which we want to emphasize.…”