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Michel Rouleux

Publications and source records attributed to Michel Rouleux.

27 records · Page 2Linked to original sources
arXiv2016

Semi-classical resonances associated with a periodic orbit of hyperbolic type

We consider in this Note resonances for a $h$-Pseudo-Differential Operator $H(x,hD_x;h)$ on $L^2(M)$ induced by a periodic orbit of hyperbolic type, as arises for Schrödinger operator with AC Stark effect when $M={\bf R}^n$, or the geodesic flow on an axially symmetric manifold $M$, extending Poincaré example of Lagrangian systems with 2 degrees of freedom. We generalize the framework of [GéSj], in the sense that we allow for hyperbolic and elliptic eigenvalues of Poincaré map, and look for so-called semi-excited resonances with imaginary part of magnitude $-h\log h$, or $h^s$, with $0<s<1$.

math-ph↗
arXiv2015

Quantum vorticity at thermal equilibrium for spins systems with continuous symmetry

We propose a definition of vorticity at inverse temperature βfor Gibbs states in quantum XY spin systems on the lattice by testing \exp[-βH] on a complete set of observables ("one-point functions"). We show in particular that it is independent of the choice of a particular basis. Imposing a compression of Pauli matrices at the boudary, which stands for the classical environment, we make some numerical simulations on finite lattices, and exhibit usual vortex patterns.

math-ph↗
arXiv2014

Multidimensional tunneling between potential wells at non degenerate minima

We consider tunneling between symmetric wells for a 2-D semi-classical Schrödinger operator for energies close to the quadratic minimum of the potential V in two cases: (1) excitations of the lowest frequency in the harmonic oscillator approximation of V; (2) more general excited states from Diophantine tori with comparable quantum numbers.

math-ph↗
arXiv2013

The semiclassical Maupertuis-Jacobi correspondence for quasi-periodic Hamiltonian flows: stable and unstable spectra

We investigate semi-classical properties of Maupertuis-Jacobi correspondence in 2-D for families of Hamiltonians $(H_λ(x,ξ), {\cal H}_λ(x,ξ))$, when ${\cal H}_λ(x,ξ)$ is the perturbation of completely integrable Hamiltonian $\widetilde{\cal H}$ veriying some isoenergetic non-degeneracy conditions. Assuming the Weyl $h$-PDO $H^w_λ$ has only discrete spectrum near $E$, and the energy surface $\{\widetilde{\cal H}={\cal E}\}$ is separated by some pairwise disjoint lagrangian tori, we show that most of eigenvalues for $\hat H_λ$ near $E$ are asymptotically degenerate as $h\to0$. This applies in particular for the determination of trapped modes by an island, in the linear theory of water-waves. We also consider quasi-modes localized near rational tori. Finally, we discuss breaking of Maupertuis-Jacobi correspondence on the equator of Katok sphere.

math-ph↗
arXiv2012

The semi-classical Maupertuis-Jacobi correspondence: stable and unstable spectra

We investigate semi-classical properties of Maupertuis-Jacobi correspondence for families of 2-D Hamiltonians $(H_λ(x,ξ), {\cal H}_λ(x,ξ))$, when ${\cal H}_λ(x,ξ)$ is the perturbation of a completely integrable Hamiltonian $\widetilde{\cal H}$ veriying some isoenergetic non-degeneracy conditions. Assuming $\hat H_λ$ has only discrete spectrum near $E$, and the energy surface $\{\widetilde{\cal H}_0={\cal E}\}$ is separated by some pairwise disjoint Lagrangian tori, we show that most of eigenvalues for $\hat H_λ$ near $E$ are asymptotically degenerate as $h\to0$. This applies in particular for the determination of trapped modes by an island, in the linear theory of water-waves. We also consider quasi-modes localized near rational tori. Finally, we discuss breaking of Maupertuis-Jacobi correspondence on the equator of Katok sphere.

math-ph↗
arXiv2008

The Semi Classical Maupertuis-Jacobi Correspondance for Quasi-Periodic Hamiltonian Flows

We extend to the semi-classical setting the Maupertuis-Jacobi correspondance for a pair of hamiltonians $(H(x,hD_x), {\cal H}(x,hD_x)$. If ${\cal H}(p,x)$ is completely integrable, or has merely has invariant diohantine torus $Λ$ in energy surface ${\cal E}$, then we can construct a family of quasi-modes for $H(x,hD_x)$ at the corresponding energy $E$. This applies in particular to the theory of water-waves in shallow water, and determines trapped modes by an island, from the knowledge of Liouville metrics.

math-ph↗
arXiv2007

Vortices and Magnetization in Kac's Model

We consider a 2-dimensional planar rotator on a large, but finite lattice with a ferromagnetic Kac potential $J_γ(i)=γ^2J(γi)$, $J$ with compact support. The system is subject to boundary conditions with vorticity. Using a Glauber like dynamics, we compute minimizers of the free energy functional at low temperature, i.e. in the regime of phase transition. We have the numerical evidence of a vortex structure for minimizers, which present many common features with those of the Ginzburg-Landau functional.

math-ph↗