Alain Joye

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

Abstract: 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 , for some C and γ >0.

Exponential decay and geometric aspect of transition probabilities in the adiabatic limit

We consider a quantum mechanical system whose hamiltonian is a time-dependent analytic 11 x n matrix. For n = 2 we establish a generalization of Dykhne formula which gives the transition probability from one energy level to the other in the adiabatic limit. We discuss in particular the geometric nature of this formula. In the general case. n > 2, we prove an upper bound for the probability of such transitions which shows that they are exponentially small. ( 1991 Academic Press. Inc

Spectral Analysis of Unitary Band Matrices

Abstract: This paper is devoted to the spectral properties of a class of unitary operators with a matrix representation displaying a band structure. Such band matrices appear as monodromy operators in the study of certain quantum dynamical systems. These doubly infinite matrices essentially depend on an infinite sequence of phases which govern their spectral properties. We prove the spectrum is purely singular for random phases and purely absolutely continuous in case they provide the doubly infinite matrix with a periodic structure in the diagonal direction. We also study some properties of the singular spectrum of such matrices considered as infinite in one direction only.

Proof of the Landau–Zener formula

Joye, A, Proof of the Landau-Zener formula, Asymptotic Analysis 9 (1994) 209-258. 209 We consider the time dependent Schrodinger equation in the adiabatic limit when the Hamiltonian is an analytic unbounded operator. It is assumed that the Hamiltonian possesses for any time two instantaneous non-degenerate eigenvalues which display an avoided crossing of finite minimum gap. We prove that the probability of a quantum transition between these two non-degenerate eigenvalues is given in the adiabatic limit by the well-known Landau-Zener formula.

Exponentially accurate semiclassical dynamics : Propagation, localization, Ehrenfest times, scattering, and more general states

Abstract. 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

Semiclassical Dynamics¶with Exponentially Small Error Estimates

\hbar

Exponentially small adiabatic invariant for the Schrödinger equation

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.

MOLECULAR PROPAGATION THROUGH SMALL AVOIDED CROSSINGS OF ELECTRON ENERGY LEVELS

Abstract: We construct approximate solutions to the time–dependent Schrödinger¶equation for small values of ħ. If V satisfies appropriate analyticity and growth hypotheses and , these solutions agree with exact solutions up to errors whose norms are bounded by for some C and γ>0. Under more restrictive hypotheses, we prove that for sufficiently small T′, implies the norms of the errors are bounded by for some C′, γ′>0, and σ > 0.

Dynamical Localization for Unitary Anderson Models

We study an adiabatic invariant for the time-dependent Schrödinger equation which gives the transition probability across a gap from timet′ to timet. When the hamiltonian depends analytically on time, andt′=−∞,t=+∞ we give sufficient conditions so that this adiabatic invariant tends to zero exponentially fast in the adiabatic limit.

General Adiabatic Evolution with a Gap Condition

This is the second of two papers on the propagation of molecular wave packets through avoided crossings of electronic energy levels in a limit where the gap between the levels shrinks as the nuclear masses are increased. An earlier paper deals with the simplest two types of generic, minimal multiplicity avoided crossings, in which the levels essentially depend on only one of the nuclear configuration parameters. The present paper deals with propagation through the remaining four types of generic, minimal multiplicity avoided crossings, in which the levels depend on more than one nuclear configuration parameter.