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

Abstract

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) ,

Non-autonomous quantum systems with scale-dependent interface conditions

“…in [2], [7], [3]). Using specific spectral assumptions, a small-θ expansion for the modified dynamical system e −itH h θ , similar to the one given in [9], has been obtained in this case (see Theorem 4.4 in [10]); nevertheless, when both h and θ are small, the resulting error term is small uniformly in time only for initial states belonging to an appropriate subspace with prescribed energy conditions. This prevents to extend the result to the most relevant case of time dependent potentials.…”

“…This yields an uniform-in-time estimate for the 'distance' between the two dynamics according to the expansion: e −itH 1 θ = e −itH 1 0 + O (|θ|), where O (·) is intended in the L 2 -operator norm sense (see theorem 1.2 in [9]). The case of h-dependent models is considered in [10] under the asymptotic regime of quantum wells in a semiclassical island; this particular framework, realized with a potential formed by the superposition of a potential barrier supported on [a, b] and potential wells supported in (a, b) with support of size h ∼ 0, has been shown to be relevant for the modelling of quantum transport systems where the charge careers couples with quasi-stationary quantum states (see e.g. in [2], [7], [3]).…”

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

“…A detailed analysis concerned with operators of this class has been developed (in a slightly more general framework) in [8]- [15]. In what follows, we resume their main features.…”

“…In a slightly more general framework, a detailed analysis of this problem has been developed in [15], where the modified dynamics is defined through a similarity, between H h θ and the corresponding self-adjoint model H h 0 , holding in some spectral subspace under the assumption that the parameters θ and h are small.…”

“…The generalized eigenfunctions of H h θ , next denoted with ψ h θ (·, k, V h ), solve of the boundary value problem (we refer to [14]- [15])…”