An explicit model for the adiabatic evolution of quantum observables driven by 1D shape resonances

Journal of Mathematical Physics

2014

Articles you may be interested inThe electronic properties of a two-electron multi-shell quantum dot-quantum well heterostructure Dispersion equation and eigenvalues for quantum wells using spectral parameter power series J. Math. Phys. 52, 043522 (2011); 10.1063/1.3579991Microcavity effect on the nonlinear intersubband response of multiple-quantum-well structures: The strongcoupling-regime Intersubband resonant enhancement of the nonlinear optical properties in compositionally asymmetric and interdiffused quantum wellsWe introduce a modified Schrödinger operator where the semiclassical Laplacian is perturbed by artificial interface conditions occurring at the boundaries of the potential's support. The corresponding dynamics is analyzed in the regime of quantum wells in a semiclassical island. Under a suitable energy constraint for the initial states, we show that the time propagator is stable with respect to the non-self-adjont perturbation, provided that this is parametrized through infinitesimal functions of the semiclassical parameter "h." It has been recently shown that h-dependent artificial interface conditions allow a new approach to the adiabatic evolution problem for the shape resonances in models of resonant heterostructures. Our aim is to provide with a rigorous justification of this method. C 2014 AIP Publishing LLC. II. A SEMICLASSICAL ISLAND WITH NON-MIXED INTERFACE CONDITIONSWe start considering the family of modified Schrödinger operators Q h θ 1 ,θ 2 (V), depending on the parameters h > 0, (θ 1 , θ 2 ) ∈ C 2 and on a potential V which is assumed to be self-adjoint and compactly supported on the bounded interval [a, b]. In particular, we set(2.1)The parameters θ 1 and θ 2 fix the interface conditions,− θ 1 2 u(a − ) = u(a + ) , e − θ 2 2 u (a − ) = u(a + ) , 4 i, j=1 B θ 1 ,θ 2 q(z, V, h) − A h θ 1 ,θ 2 −1 B θ 1 ,θ 2 i j × γz ,h (e j , V), · L 2 (R) γ z,h (e i , V) ,

Section: )mentioning

confidence: 99%

“…possibly depending on h, fulfilling for any ψ, ϕ ∈ D(Q h (V)) the equation 9) and such that the transformation (Γ 0 , Γ 1 ) :…”

Section: Boundary Triples and Krein-like Resolvent Formulasmentioning

confidence: 99%

Quantum evolution in the regime of quantum wells in a semiclassical island with artificial interface conditions

Journal of Mathematical Physics

2014

“…for a suitable N 0 ∈ N; an explicit example of this mechanism can be found in [13]. Studying the Green function around a resonant energy requires the introduction of a Dirichlet problem in order to resolve the spectral singularity and to match the complete problem with some combination of this spectral problem with the filled wells spectral problem.…”

Section: Generalized Eigenfunctions Expansionmentioning

confidence: 99%

Wave operators, similarity and dynamics for a class of Schrödinger operators with generic non-mixed interface conditions in 1D

Journal of Mathematical Physics

2013

We consider a simple modification of the 1D-Laplacian where non-mixed interface conditions occur at the boundaries of a finite interval. It has recently been shown that Schrödinger operators having this form allow a new approach to the transverse quantum transport through resonant heterostructures. In this perspective, it is important to control the deformations effects introduced on the spectrum and on the time propagator by this class of non-selfadjont perturbations. In order to obtain uniform-in-time estimates of the perturbed semigroup, our strategy consists in constructing stationary waves operators allowing to intertwine the modified non-selfadjoint Schrödinger operator with a 'physical' Hamiltonian. For small values of a deformation parameter 'θ', this yields a dynamical comparison between the two models showing that the distance between the corresponding semigroups is dominated by |θ| uniformly in time in the L 2 -operator norm.

“…Nevertheless, assuming ε = e −τ /h for some τ > 0, it is still possible to obtain an error bound exponentially small w.r.t. h. Next, we recall, in the simplified case of a single shape resonance, the result of the adiabatic theorem provided with in [8,Theorem 7.1] (see also [9] for the explicit form of the modulation factor µ (t) below). Theorem 2.3.…”

Section: Under These Assumptions Hmentioning

confidence: 99%

“…In [9], the small-h behavior of the solution of (56) has been investigated for a time dependent potential formed by a flat barrier of height V 0 > 0 plus an attractive, timedependent delta interaction acting in x 0 ∈ (a, b) and preserving the quantum-well scaling. Explicitly, we consider the model:…”

Section: An Explicit Example and Final Remarksmentioning

confidence: 99%

A linearized model of quantum transport in the asymptotic regime of quantum wells

Nanosist.: fiz. him. mat.

2015

The effects of the local accumulation of charges in resonant tunnelling heterostructures have been described using 1D Shrödinger-Poisson Hamiltonians in the asymptotic regime of quantum wells. Taking into account the features of the underling physical system, the corresponding linearized model is naturally related to the adiabatic evolution of shape resonances on a time scale which is exponentially large w.r.t. the asymptotic parameter h. A possible strategy to investigate this problem consists of using a complex dilation to identify the resonances with the eigenvalues of a deformed operator. Then, the adiabatic evolution problem for a sheet-density of charges can be reformulated using the deformed dynamical system which, under suitable initial conditions, is expected to evolve following the instantaneous resonant states. After recalling the main technical difficulties related to this approach, we introduce a modified model where h-dependent artificial interface conditions, occurring at the boundary of the interaction region, allow one to obtain adiabatic approximations for the relevant resonant states, while producing a small perturbation of the dynamics on the scale h N0 . According to these results, we finally suggest an alternative formulation of the adiabatic problem. An a posteriori justification of our method is obtained by considering an explicitlysolvable case.