QMath9 Abstracts

Abstracts of Quantum Chaos and Semiclassics section

Bolte, Jens Quantum Chaos and Semiclassics Semiclassical propagation of coherent states with spin-orbit interaction We study semiclassical approximations to the time evolution of coherent states for general spin-orbit coupling problems in two different semiclassical scenarios: The limit $\hbar\to 0$ is first taken with fixed spin quantum number $s$ and then with $\hbar s$ held constant. In these two cases different classical spin-orbit dynamics emerge. We prove that a coherent state propagated with a suitable classical dynamics approximates the quantum time evolution up to an error of size $\sqrt{\hbar}$ and identify an Ehrenfest time scale. Subsequently an improvement of the semiclassical error to an arbitray order $\hbar^{N/2}$ is achieved by a suitable deformation of the state that is propagated classically.
De Bièvre, Stephan Quantum Chaos and Semiclassics The semi-classical behaviour of quantum map eigenfunctions Quantum maps are a theoretical laboratory for questions in quantum chaos. I will discuss here some recent results in the understanding of the eigenfunction behaviour of such systems in the semi-classical limit.
Fournais, Soeren Quantum Chaos and Semiclassics On the ground state of magnetic hamiltonians Joint with B. Helffer. In this talk I will present recent results on the asymptotics of the ground state of the Neumann realisation of the usual magnetic Schrodinger operator in a two-dimensional domain. By a procedure of reduction to the boundary we obtain a complete asymptotics, in the semiclassical regime, of the lowest lying eigenvalues. Due to applications to super-conductivity this problem has been studied a lot recently in the litterature. We mention the works by Bauman-Phillips-Tang, Lu-Pan, del Pino-Felmer- Sternberg and Helffer-Morame. Our results are a sharpening of these previous results.
Iantchenko, Alexei Quantum Chaos and Semiclassics Resonances associated to trapped hyperbolic trajectory
Julio Hugo Toloza, Julio Quantum Chaos and Semiclassics Exponentially accurate quasimodes for the time-independent Born-Oppenheimer approximation
Keppeler, Stefan Quantum Chaos and Semiclassics Semiclassical quantisation rules for the Dirac and Pauli equations We derive explicit semiclassical quantisation conditions for the Dirac and Pauli equations [1,2]. We show that the spin degree of freedom yields a contribution, deriving from a non-Abelian geometric phase, which is of the same order of magnitude as the Maslov correction in Einstein-Brillouin-Keller quantisation. In order to obtain this result a generalisation of the notion of integrability for a certain skew product flow of classical translational dynamics and classical spin precession has to be derived. As an example we discuss the relativistic Kepler problem with Thomas precession, whose treatment sheds some light on the amazing success of Sommerfeld's theory of fine structure [3]. The approach generalises to other multi-component wave equations. [1] S. Keppeler, Phys. Rev. Lett. 89 (2002) 210405 and Ann. Phys. (NY) 304 (2003) 40-71 [2] S. Keppeler, Spinning Particles: Semiclassics and Spectral Statistics, Spinger Tracts in Modern Physics, vol. 193, Springer-Verlag Berlin Heidelberg, 2003 [3] A. Sommerfeld, Ann. Phys. (Leipzig) 51 (1916) 1-94 125-167
Lev, Ondrej Quantum Chaos and Semiclassics Asympotic behavior of some matrix elements
Marklof, Jens Quantum Chaos and Semiclassics Weyl's law and quantum ergodicity for maps with divided phase space For a general class of unitary quantum maps, whose underlying classical phase space is divided into ergodic and non-ergodic components, we prove analogues of Weyl's law for the distribution of eigenphases, and the Schnirelman-Zelditch-Colin de Verdiere Theorem on the equidistribution of eigenfunctions with respect to the ergodic components of the classical map (quantum ergodicity). We apply our main theorems to quantised linked twist maps on the torus.
Muno, Ralf Quantum Chaos and Semiclassics
NONNENMACHER, Stéphane Quantum Chaos and Semiclassics Long-time evolution for quantized Anosov maps
Nicoleau, François Quantum Chaos and Semiclassics {Inverse scattering for a Schr\" odinger operator with a repulsive potential We We consider a pair of Hamiltonians $(H, H_0 )$ on $L^2 (\r^n )$ where $H_0= p^2 -x^2$ is a Schr\" odinger operator with a repulsive potential, and $H= H_0 +V(x)$. We show that, under suitable assumptions on the decay of the electric potential, $V$ is uniquely determined by the high energy limit of the scattering operator.
PAUL, Thierry Quantum Chaos and Semiclassics Long time semiclassical asymptotics We present some results on the long time semiclassical evolution for wave packets and observables-both geometrical and microlocal aspects are discussed. Some related results concerning quantum hydrodynamics and regularization of hyperbolic type non linear partial differential equations in the presence of shocks are also presented. Moreover semiclassical methods are used to study the quantum evolution for complex hamiltonians and the dispersionless (large number of particles) limit of (classical) Toda system.
Rouleux, Michel Quantum Chaos and Semiclassics Instantons and tunnel cycles We present some geometrical technics to compute instantons and semiclassical tunneling between multidimensional tori in phase space ; these apply for double wells in case of the Schr\"odinger operator $P=-h^2\Delta+V(x)$, or the magnetic Schr\"odinger operator $P=(hD-A(x))^2+V(x)$, or the Laplace-Beltrami operator on a 2-d Liouville surface.
Rudnick, Zeev Quantum Chaos and Semiclassics Eigenvalue statistics
Schubert, Roman Quantum Chaos and Semiclassics Time evolution of wave packets for large times We study the semiclassical propagation of wave packets in quantum mechanics for large times. The results depend strongly on the ergodic properties of the underlying classical system. If it is Anosov we obtain universal equidistribution for the class of wavepackets we consider, whereas for classically integrable the behavior depends strongly on the initial conditions.
Soshnikov, Alexander Quantum Chaos and Semiclassics Poisson Statistics for the Largest Eigenvalues of Random Matrices with Heavy Tails We study large Wigner random matrices and sample covariance random matrices in the case when the matrix entries have distributions with heavy tails. We prove that the largest eigenvalues of such random matrices have Poisson statistics in the limit.
Villegas-Blas, Carlos Quantum Chaos and Semiclassics The Bargmann Transform, the Hopf fibration and regularization of the n=2,3,5 dimensional Kepler problem We introduce a Bargmann transform for the space $L^2(S^n)$ of square integrable functions on the $n=2,3,5$ dimensional unit sphere $S^n$ inmersed in ${\mathbb R}^{n+1}$. This is done on base of the Hopf fibration for the spheres $S^K\mapsto{S}^d$ with $(k,d)=(1,1),$ $(3,2),$ $(7,4)$ and a suitable canonical transformation relating two different ways to regularize the $n=2,3,5$ dimensional Kepler problem (for negative energy) involving the null quadrics in ${\mathbb C}^m$, $m=3,4,6$.The unitarity of the Bargmann transform onto a suitable space of analytical functions is showed. We also show reproducing kernels for these spaces of analytical functions. Since our Bargmann transform is actually a coherent states transform we provide sets of coherent states for both $L^2(S^n)$, $n=2,3,5$ and the negative energy Hilbert space of the $n=2,3,5$ dimensional hydrogen atom problem.
Zworski, Maciej Quantum Chaos and Semiclassics TBA