Thierry Paul

Transforms associated to square integrable group representations. I. General results

Let G be a locally compact group, which need not be unimodular. Let x→U(x) (x∈G) be an irreducible unitary representation of G in a Hilbert space H(U). Assume that U is square integrable, i.e., that there exists in H(U) at least one nonzero vector g such that ∫‖(U(x)g,g)‖2 dx<∞. We give here a reasonably self‐contained analysis of the correspondence associating to every vector f∈H(U) the function (U(x)g,f) on G, discussing its isometry, characterization of the range, inversion, and simplest interpolation properties. This correspondence underlies many properties of generalized coherent states.

Functions analytic on the half-plane as quantum mechanical states

A transform between the state space of one‐dimensional quantum mechanical systems and a direct sum of two spaces of square integrable functions analytic on the open upper half‐plane is constructed. It gives a representation of usual quantum mechanics on which the free evolution is trivial and the representation of canonical transformation very simple. Generalizations to higher dimensions are also discussed.

Wiener measures for path integrals with affine kinematic variables

The results obtained earlier have been generalized to show that the path integral for the affine coherent state matrix element of a unitary evolution operator exp(−iTH) can be written as a well‐defined Wiener integral, involving Wiener measure on the Lobachevsky half‐plane, in the limit that the diffusion constant diverges. This approach works for a wide class of Hamiltonians, including, e.g., −d2/dx2+V(x) on L2(R+), with V sufficiently singular at x=0.

The Schrödinger equation and canonical perturbation theory

LetT0(ħ, ω)+εV be the Schrödinger operator corresponding to the classical HamiltonianH0(ω)+εV, whereH0(ω) is thed-dimensional harmonic oscillator with non-resonant frequencies ω=(ω1, ... , ωd) and the potentialV(q1, ... ,qd) is an entire function of order (d+1)−1. We prove that the algorithm of classical, canonical perturbation theory can be applied to the Schrödinger equation in the Bargmann representation. As a consequence, each term of the Rayleigh-Schrödinger series near any eigenvalue ofT0(ħ, ω) admits a convergent expansion in powers of ħ of initial point the corresponding term of the classical Birkhoff expansion. Moreover ifV is an even polynomial, the above result and the KAM theorem show that all eigenvalues λn(ħ, ε) ofT0+εV such thatnħ coincides with a KAM torus are given, up to order ε∞, by a quantization formula which reduces to the Bohr-Sommerfeld one up to first order terms in ħ.

Phasing in and Phasing Out Protectionism with Costly Adjustment of Labour

We study the dynamics of optimal trade policy in a model with costly inter-sectoral adjustment of labour, where migrants pay less than the marginal social cost of migration. If workers have rational expectations, a future tariff has an announcement effect on the current migration decision. If the government is able to commit itself to future policy, the optimal trajectory involves phasing in and then phasing out protection of the dying sector. This contrasts with recommendations of gradual liberalization. Without the ability to make commitments, the equilibrium policy begins with and maintains free trade.

Dispersionless Toda and Toeplitz operators

In this paper we present some results on the dispersionless limit of the Toda lattice equations viewed as the semiclassical limit of an equation involving certain Toeplitz operators. We consider both nonperiodic and periodic boundary conditions. For the nonperiodic case the phase space is the Riemann sphere, while in the periodic case it is the torusC/Z2. In both cases we prove precise estimates on the dispersionless limit. In addition, we show that the Toda equations, although they are nonlinear, propagate a Toeplitz operator into an operator arbitrarily close to a Toeplitz operator as long as the Toda partial differential equation (PDE) (dispersionless limit) admits smooth solutions.

Coarse-scale representations and smoothed Wigner transforms

Abstract Smoothed Wigner transforms have been used in signal processing, as a regularized version of the Wigner transform, and have been proposed as an alternative to it in the homogenization and/or semiclassical limits of wave equations. We derive explicit, closed formulations for the coarse-scale representation of the action of pseudodifferential operators. The resulting “smoothed operators” are in general of infinite order. The formulation of an appropriate framework, resembling the Gelfand–Shilov spaces, is necessary. Similarly we treat the “smoothed Wigner calculus”. In particular this allows us to reformulate any linear equation, as well as certain nonlinear ones (e.g., Hartree and cubic nonlinear Schrodinger), as coarse-scale phase-space equations (e.g., smoothed Vlasov), with spatial and spectral resolutions controlled by two free parameters. Finally, it is seen that the smoothed Wigner calculus can be approximated, uniformly on phase-space, by differential operators in the semiclassical regime. This improves the respective weak-topology approximation result for the Wigner calculus.

Strong semiclassical approximation of Wigner functions for the Hartree dynamics

A. Athanassoulis would like to thank the CMLS, Ecole polytechnique for its hospitality during the preparation of this work. He was also partially supported by Award No. KUK-I1-007-43, funded by the King Abdullah University of Science and Technology (KAUST). F. Pezzotti was partially supported by Project CBDif-Fr ANR-08-BLAN-0333-01.

On the Mean Field and Classical Limits of Quantum Mechanics

The main result in this paper is a new inequality bearing on solutions of the N-body linear Schrödinger equation and of the mean field Hartree equation. This inequality implies that the mean field limit of the quantum mechanics of N identical particles is uniform in the classical limit and provides a quantitative estimate of the quality of the approximation. This result applies to the case of C1,1 interaction potentials. The quantity measuring the approximation of the N-body quantum dynamics by its mean field limit is analogous to the Monge–Kantorovich (or Wasserstein) distance with exponent 2. The inequality satisfied by this quantity is reminiscent of the work of Dobrushin on the mean field limit in classical mechanics [Func. Anal. Appl. 13, 115–123, (1979)]. Our approach to this problem is based on a direct analysis of the N-particle Liouville equation, and avoids using techniques based on the BBGKY hierarchy or on second quantization.

Strong phase-space semiclassical asymptotics

Wigner and Husimi transforms have long been used for the phase-space reformulation of Schrodinger-type equations and the study of the corresponding semiclassical limits. Most of the existing results provide approximations in appropriate weak topologies. In this work we are concerned with semiclassical limits in the strong topology, i.e., approximation of Wigner functions by solutions of the Liouville equation in