arXiv subjects

Andrea Mantile

Publications and source records attributed to Andrea Mantile.

27 records · Page 2Linked to original sources

arXiv2015

A linearized model of quantum transport with interface conditions in the adiabatic regime

We introduce a modified model where h-dependent artificial interface conditions, occurring at the boundary of an interaction region, allow to obtain adiabatic approximations for the relevant resonant states connected to the quantum transport problem. Under positivity assumptions on the potential, we show that this modification produces a small perturbation of the dynamics on any time scale. The result extends to the corresponding non-autonomous case, allowing us to consider the adiabatic evolution problem in the modified setting. In this framework, we give an expansion formula for the small-h asymptotics of the adiabatic variable, where the error introduced on the adiabatic dynamics by the interface conditions is polynomially small w.r.t. h, independently from the adiabatic time scale. This result provides with a rigorous mathematical framework for the interface conditions approach to the analysis of the adiabatic transport problem in the quantum wells regime.

math-ph↗

arXiv2015

Non-autonomous quantum systems with scale-dependent interface conditions

We consider a class of modified Schroedinger operators where the semiclassical Laplacian is perturbed with h-dependent interface conditions occurring at the boundaries of the potential's support. Under positivity assumptions on the potential, we show that this modification produces a small perturbation on the dynamics as h->0, independently from the time scale. In the case of a time dependent potential, this yields uniform-in-h stability estimates for products of instantaneous propagators. Then, following a standard approach, the non-autonomous dynamical system is defined as a limit of stepwise propagators and its small-h expansion is provided under suitable regularity assumptions on the potential's variations.

math-ph↗

arXiv2013

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

We 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 analysed 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 w.r.t. the non-selfadjont 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.

math-ph↗

arXiv2013

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

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.

math-ph↗

arXiv2010

Stability and control of a 1D quantum system with confining time dependent delta potentials

The evolution problem for a quantum particle confined in a 1D box and interacting with one fixed point through a time dependent point interaction is considered. Under suitable assumptions of regularity for the time profile of the Hamiltonian, we prove the existence of strict solutions to the corresponding Schrödinger equation. The result is used to discuss the stability and the steady-state local controllability of the wavefunction when the strenght of the interaction is used as a control parameter.

math.AP↗

arXiv2010

Adiabatic evolution of 1D shape resonances: an artificial interface conditions approach

Artificial interface conditions parametrized by a complex number $θ_{0}$ are introduced for 1D-Schrödinger operators. When this complex parameter equals the parameter $θ\in i\R$ of the complex deformation which unveils the shape resonances, the Hamiltonian becomes dissipative. This makes possible an adiabatic theory for the time evolution of resonant states for arbitrarily large time scales. The effect of the artificial interface conditions on the important stationary quantities involved in quantum transport models is also checked to be as small as wanted, in the polynomial scale $(h^N)_{N\in \N}$ as $h\to 0$, according to $θ_{0}$.

math.AP↗

arXiv2009

Time-dependent delta-interactions for 1D Schrödinger Hamiltonians

The non autonomous Cauchy problem for time dependent 1D point interactions is considered. The regularity assumptions for the coupling parameter are accurately analyzed and show that the general results for non autonomous linear evolution equations in Banach spaces are far from being optimal. In the mean time, this article shows an unexpected application of paraproduct techniques, initiated by J.M. Bony for nonlinear partial differential equations, to a classical linear problem.

math-ph↗

arXiv2007

On the optimization of the principal eigenvalue for single-centre point-interaction operators in a bounded region

We investigate relations between spectral properties of a single-centre point-interaction Hamiltonian describing a particle confined to a bounded domain $Ω\subset\mathbb{R}^{d},\: d=2,3$, with Dirichlet boundary, and the geometry of $Ω$. For this class of operators Krein's formula yields an explicit representation of the resolvent in terms of the integral kernel of the unperturbed one, $(-Δ_Ω^{D}+z) ^{-1}$. We use a moving plane analysis to characterize the behaviour of the ground-state energy of the Hamiltonian with respect to the point-interaction position and the shape of $Ω$, in particular, we establish some conditions showing how to place the interaction to optimize the principal eigenvalue.

math-ph↗

arXiv2007

Point Interaction Controls for the Energy Transfer in 3-D Quantum Systems

We consider the problem of energy-mass transfer from scattering to bound states for a one body quantum system under the action of a time dependent point interaction. Under suitable assumptions on the initial state of the particle, we prove a result of local controllability of this process

math.OC↗