André Martinez

An introduction to semiclassical and microlocal analysis

Introduction * Semiclassical Pseudodifferential Calculus * Microlocalization * Applications to the Solutions of Analytic Linear PDEs * Complements: Symplectic Aspects * Appendix: List of Formulae * Bibliography * Index * List of Notations

On the Born-Oppenheimer Expansion for Polyatomic Molecules

We consider the Schrödinger operatorP(h) for a polyatomic molecule in the semiclassical limit where the mass ratioh2 of electronic to nuclear mass tends to zero. We obtain WKB-type expansions of eigenvalues and eigenfunctions ofP(h) to all orders inh. This allows to treat the splitting of the ground state energy of a non-planar molecule. Our class of potentials covers the physical case of the Coulomb interaction. We use methods ofh-pseudodifferential operators with operator valued symbols, which by use of appropriate coordinate changes in local coordinate patches covering the classically accessible region become applicable even to our class of singular potentials.

Ergodicité et limite semi-classique

Consider a self adjoint quantic hamiltonian:P(h)=p(x, hDx) whereh>0 is the Plancks constant andp some smooth classical observable on the phase space R2n. When the classical flow on a compact energy shell {p=λ} is ergodic we prove that in the limith ↓ 0 almost all the eigenfunctions ofP(h) whose energy is near of λ are distributed according to the Liouville measure on {p=λ}.In the high energy case (λ →+∞) this sort of problem was considered by A. Schnirelman, S. Zelditch, and Y. Colin de Verdière.

A mathematical approach to the effective Hamiltonian in perturbed periodic problems

We describe a rigorous mathematical reduction of the spectral study for a class of periodic problems with perturbations which gives a justification of the method of effective Hamiltonians in solid state physics. We study the partial differential operators of the formP=P(hy, y, Dy+A(hy)) onRn (whenh>0 is small enough), whereP(x, y, η) is elliptic, periodic iny with respect to some lattice Γ, and admits smooth bounded coefficients in (x, y).A(x) is a magnetic potential with bounded derivatives. We show that the spectral study ofP near any fixed energy level can be reduced to the study of a finite system ofh-pseudodifferential operatorsE(x, hDx, h), acting on some Hilbert space depending on Γ. We then apply it to the study of the Schrödinger operator when the electric potential is periodic, and to some quasiperiodic potentials with vanishing magnetic field.

Twisted pseudodifferential calculus and application to the quantum evolution of molecules

We construct an abstract pseudodifferential calculus with operator-valued symbol, adapted to the treatment of Coulomb-type interactions, and we apply it to study the quantum evolution of molecules in the Born-Oppenheimer approximation, in the case where the electronic Hamiltonian admits a local gap in its spectrum. In particular, we show that the molecular evolution can be reduced to the one of a system of smooth semiclassical operators, the symbol of which can be computed explicitely. In addition, we study the propagation of certain wave packets up to long time values of Ehrenfest order. (This work has been accepted for publication as part of the Memoirs of the American Mathematical Society and will be published in a future volume.)

Precise exponential estimates in adiabatic theory

General adiabatic evolutions associated to Hamiltonians, which admit a holomorphic extension with respect to the time variable in a complex strip, and whose spectrum satisfies a gap condition are studied. An explicit rate of exponential decay is given, which is related to simple geometric quantities associated to the spectrum of the Hamiltonian, for the transition probability between the two parts of the spectrum when the evolution is taken from −∞ to +∞.

MEAN-FIELD APPROXIMATION OF QUANTUM SYSTEMS AND CLASSICAL LIMIT

We prove that, for a smooth two-body potentials, the quantum mean-field approximation to the nonlinear Schroedinger equation of the Hartree type is stable at the classical limit h \to 0, yielding the classical Vlasov equation.

On the Born-Oppenheimer approximation of wave operators in molecular scattering theory

In this paper we study the diatomic molecular scattering by reducing the number of particles through Born-Oppenheimer approximation. Under a non-trapping assumption on the effective potential of the molecular Hamiltonian we use semiclassical resolvent estimates to show that non-adiabatic corrections to the adiabatic (or Born-Oppenheimer) wave operators are small. Furthermore we study the classical limit of the adiabatic wave operators by computing its action on quantum observables microlocalized by use of coherent states.

Breit-Wigner formulas for the scattering phase and the total scattering cross-section in the semi-classical limit

In this paper we prove results in resonance scattering for the Schrödinger operatorPv=−h2Δ+V, V being a smooth, short range potential onRn. More precisely, for energy λ near a trapping energy level λ0 for the classical system defined by the Hamiltonianp(x,ζ)=ζ2+V(x), we prove that the scattering phase and the scattering cross sections associated to (Pv, P0) have the Breit-Wigner form (“Lorentzian line shape”) in the limith→0.

Resonances of diatomic molecules in the Born-Oppenheimer approximation

We study the spectral properties of general diatomic molecules near a scattering level. We prove that in the Born-Oppenheimer approximation, this study can be reduced to the one of a a matrix of semiclassical pseudodifferential operators. More precisely, we provide a link between the resonances of the molecule which are close enough to the real axis, and the discrete spectrum of this matrix (the principal symbol of which being made explicit via the electronic levels of the molecule).