arXiv subjects

Alain Joye

Publications and source records attributed to Alain Joye.

61 records · Page 4Linked to original sources

arXiv2000

Avoided crossings in mesoscopic systems: electron propagation on a non-uniform magnetic cylinder

We consider an electron constrained to move on a surface with revolution symmetry in the presence of a constant magnetic field $B$ parallel to the surface axis. Depending on $B$ and the surface geometry the transverse part of the spectrum typically exhibits many crossings which change to avoided crossings if a weak symmetry breaking interaction is introduced. We study the effect of such perturbations on the quantum propagation. This problem admits a natural reformulation to which tools from molecular dynamics can be applied. In turn, this leads to the study of a perturbation theory for the time dependent Born-Oppenheimer approximation.

math-ph↗

arXiv2000

A Time-Dependent Born-Oppenheimer Approximation with Exponentially Small Error Estimates

We present the construction of an exponentially accurate time-dependent Born-Oppenheimer approximation for molecular quantum mechanics. We study molecular systems whose electron masses are held fixed and whose nuclear masses are proportional to $ε^{-4}$, where $ε$ is a small expansion parameter. By optimal truncation of an asymptotic expansion, we construct approximate solutions to the time-dependent Schrödinger equation that agree with exact normalized solutions up to errors whose norms are bounded by $\ds C \exp(-γ/ε^2)$, for some C and $γ>0$.

math-ph↗

arXiv1999

Exponentially Accurate Semiclassical Dynamics: Propagation, Localization, Ehrenfest Times, Scattering and More General States

We prove six theorems concerning exponentially accurate semiclassical quantum mechanics. Two of these theorems are known results, but have new proofs. Under appropriate hypotheses, they conclude that the exact and approximate dynamics of an initially localized wave packet agree up to exponentially small errors in $\hbar$ for finite times and for Ehrenfest times. Two other theorems state that for such times the wave packets are localized near a classical orbit up to exponentially small errors. The fifth theorem deals with infinite times and states an exponentially accurate scattering result. The sixth theorem provides extensions of the other five by allowing more general initial conditions.

math-ph↗

arXiv1999

Edge currents in the absence of edges

We investigate a charged two-dimensional particle in a homogeneous magnetic field interacting with a periodic array of point obstacles. We show that while Landau levels remain to be infinitely degenerate eigenvalues, between them the system has bands of absolutely continuous spectrum and exhibits thus a transport along the array. We also compute the band functions and the corresponding probability current.

cond-mat↗

arXiv1998

Semiclassical Dynamics with Exponentially Small Error Estimates

We construct approximate solutions to the time--dependent Schrödinger equation $i \hbar (\partial ψ)/(\partial t) = - (\hbar^2)/2 Δψ+ V ψ$ for small values of $\hbar$. If $V$ satisfies appropriate analyticity and growth hypotheses and $|t|\le T$, these solutions agree with exact solutions up to errors whose norms are bounded by $C \exp{-γ/\hbar}$, for some $C$ and $γ>0$. Under more restrictive hypotheses, we prove that for sufficiently small $T', |t|\le T' |\log(\hbar)|$ implies the norms of the errors are bounded by $C' \exp{-γ'/\hbar^σ}$, for some $C', γ'>0$, and $σ>0$.

math-ph↗

arXiv1995

Exponential Asymptotics in a Singular Limit for $n$-Level Scattering Systems

The singular limit $\eps\ra 0$ of the $S$-matrix associated with the equation $i\eps dψ(t)/dt=H(t)ψ(t)$ is considered, where the analytic generator $H(t)\in M_n(\C)$ is such that its spectrum is real and non-degenerate for all $t\in\R$. Sufficient conditions allowing to compute asymptotic formulas for the exponentially small off-diagonal elements of the $S$-matrix as $\eps\ra 0$ are explicited and a wide class of generators for which these conditions are verified is defined. These generators are obtained by means of generators whose spectrum exhibits eigenvalue crossings which are perturbed in such a way that these crossings turn to avoided crossings. The exponentially small asymptotic formulas which are derived are shown to be valid up to exponentially small relative error, by means of a joint application of the complex WKB method together with superasymptotic renormalization. The application of these results to the study of quantum adiabatic transitions in the time dependent Schrödinger equation and of the semiclassical scattering properties of the multichannel stationary Schrödinger equation closes this paper. The results presented here are a generalization to $n$-level systems, $n\geq 2$, of results previously known for $2$-level systems only.

quant-ph↗

arXiv1994

An Adiabatic Theorem for Singularly Perturbed Hamiltonians

The adiabatic approximation in quantum mechanics is considered in the case where the self-adjoint hamiltonian $H_0(t)$, satisfying the usual spectral gap assumption in this context, is perturbed by a term of the form $εH_1(t)$. Here $ε\to 0$ is the adiabaticity parameter and $H_1(t)$ is a self-adjoint operator defined on a smaller domain than the domain of $H_0(t)$. Thus the total hamiltonian $H_0(t)+εH_1(t)$ does not necessarily satisfy the gap assumption, $\forall ε>0$. It is shown that an adiabatic theorem can be proven in this situation under reasonnable hypotheses. The problem considered can also be viewed as the study of a time-dependent system coupled to a time-dependent perturbation, in the limit of large coupling constant.

funct-an↗