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Thierry Paul

Publications and source records attributed to Thierry Paul.

69 records · Page 4Linked to original sources
arXiv2010

On the propagation of oceanic waves driven by a strong macroscopic flow

In this work we study oceanic waves in a shallow water flow subject to strong wind forcing and rotation, and linearized around a inhomogeneous (non zonal) stationary profile. This extends the study \cite{CGPS}, where the profile was assumed to be zonal only and where explicit calculations were made possible due to the 1D setting. Here the diagonalization of the system, which allows to identify Rossby and Poincaré waves, is proved by an abstract semi-classical approach. The dispersion of Poincaré waves is also obtained by a more abstract and more robust method using Mourre estimates. Only some partial results however are obtained concerning the Rossby propagation, as the two dimensional setting complicates very much the study of the dynamical system.

math.AP↗
arXiv2010

Smoothed Affine Wigner Transform

We study a generalization of Husimi function in the context of wavelets. This leads to a nonnegative density on phase-space for which we compute the evolution equation corresponding to a SchrÄodinger equation.

math.AP↗
arXiv2010

Semiclassical limit of quantum dynamics with rough potentials and well posedness of transport equations with measure initial data

In this paper we study the semiclassical limit of the Schrödinger equation. Under mild regularity assumptions on the potential $U$ which include Born-Oppenheimer potential energy surfaces in molecular dynamics, we establish asymptotic validity of classical dynamics globally in space and time for "almost all" initial data, with respect to an appropriate reference measure on the space of initial data. In order to achieve this goal we prove existence, uniqueness and stability results for the flow in the space of measures induced by the continuity equation.

math.AP↗
arXiv2010

Bohmian measures and their classical limit

We consider a class of phase space measures, which naturally arise in the Bohmian interpretation of quantum mechanics (when written in a Lagrangian form). We study the so-called classical limit of these Bohmian measures, in dependence on the scale of oscillations and concentrations of the sequence of wave functions under consideration. The obtained results are consequently compared to those derived via semi-classical Wigner measures. To this end, we shall also give a connection to the, by now classical, theory of Young measures and prove several new results on Wigner measures themselves. We believe that our analysis sheds new light on the classical limit of Bohmian quantum mechanics and gives further insight on oscillation and concentration effects of semi-classical wave functions.

math-ph↗
arXiv2010

Strong phase-space semiclassical asymptotics

Wigner and Husimi transforms have long been used for the phase-space reformulation of Schrödinger-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 $L^2$ and Sobolev norms. The results obtained improve the state of the art, and highlight the role of potential regularity, especially through the regularity of the Wigner equation. It must be mentioned that the strong convergence can be shown up to $O(log \frac{1}ε)$ time-scales, which is well known to be, in general, the limit of validity of semiclassical asymptotics.

math.AP↗
arXiv2009

De Toda à KdV

We consider the large number of particles limit of a periodic Toda lattice for a family of initial data close to the equilibrium state. We show that each of the two edges of the spectra of the corresponding Jacobi matrices is up to an error, determined by the spectra of two Hill operators, associated to this family. We then show that the spectra of the Jacobi matrices remain almost constant when the matrices evolve along the two limiting KdV equations. Finally we prove that the Toda actions, when appropriately renormalized, converge to the ones of KdV .

math.AP↗
arXiv2009

Poincaré et les quanta

We present some reflections concerning two papers by H. Poincaré concerning the theory of quanta.

math.HO↗
arXiv2009

Trapping Rossby waves

Waves associated to large scale oceanic motions are gravity waves (Poincaré waves which disperse fast) and quasigeostrophic waves (Rossby waves). In this Note, we show by semiclassical arguments, that Rossby waves can be trapped and we characterize the corresponding initial conditions.

math.AP↗
arXiv2009

Semiclassical analysis and sensitivity to initial conditions

We present several recent results concerning the transition between quantum and classical mechanics, in the situation where the underlying dynamical system has an hyperbolic behaviour. The special role of invariant manifolds will be emphasized, and the long time evolution will show how the quantum non-determinism and the classical chaotic sensitivity to initial conditions can be compared, and in a certain sense overlap.

math-ph↗
arXiv2009

Some Remarks About Semiclassical Trace Invariants and Quantum Normal Forms

In this paper we explore the connection between semi-classical and quantum Birkhoff canonical forms (BCF) for Schrodinger operators. In particular we give a "non-symbolic" operator theoretic derivation of the quantum Birkhoff canonical form and provide an explicit recipe for expressing the quantum BCF in terms of the semi-classical BCF.

math-ph↗
arXiv2008

Semi-classical Analysis of Spin Systems near Critical Energies

The spectral properties of $su(2)$ Hamiltonians are studied for energies near the critical classical energy $ε_c$ for which the corresponding classical dynamics presents hyperbolic points (HP). A general method leading to an algebraic relation for eigenvalues in the vicinity of $ε_c$ is obtained in the thermodynamic limit, when the semi-classical parameter $n^{-1}=(2s)^{-1}$ goes to zero (where $s$ is the total spin of the system). Two applications of this method are given and compared with numerics. Matrix elements of observables, computed between states with energy near $ε_c$, are also computed and shown to be in agreement with the numerical results.

quant-ph↗
arXiv2008

Coarse-scale representations and smoothed Wigner transforms

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 non-linear Schrödinger), 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.

math.AP↗
arXiv2006

Remarks on Boundary Layer Expansions

A systematic mathematical methodology for derivation of boundary layer expansions is presented. An explicit calculation of boundary layer sizes is given and proved to be coordinates system independent. It relies on asymptotic properties of symbols of operators. Several examples, including the quasigeostrophic model, are discussed.

math.AP↗
arXiv1998

Long time semiclassical approximation of quantum flows: a proof of the Ehrenfest time

Let ${\cal H}(x,ξ)$ be a holomorphic Hamiltonian of quadratic growth on $ R^{2n}$, $b$ a holomorphic exponentially localized observable, $H$, $B$ the corresponding operators on $L^2(R^n)$ generated by Weyl quantization, and $U(t)=\exp{iHt/\hbar}$. It is proved that the $L^2$ norm of the difference between the Heisenberg observable $B_t=U(t)BU(-t)$ and its semiclassical approximation of order ${N-1}$ is majorized by $K N^{(6n+1)N}(-\hbar ln\hbar)^N$ for $t\in [0,T_N(\hbar)]$ where $T_N(\hbar)=-{2 ln\hbar\over {N-1}}$. Choosing a suitable $N(\hbar)$ the error is majorized by $C\hbar^{ln|ln\hbar|}$, $0\leq t\leq |ln\hbar|/ln|ln\hbar|$. (Here $K,C$ are constants independent of $N,\hbar$).

math-ph↗