arXiv subjects

Hanen Louati

Publications and source records attributed to Hanen Louati.

5 recordsLinked to original sources

arXiv2021

On semi-classical spectral series for an atom in a periodic polarized electric field

In this report we present preliminary results about the tunneling problem for a magnetic Schrödinger operator. As a motivation we consider the 3-D time-dependent Schrödinger operator $H(t)=-h^2Δ+V+E(t)\cdot x$ where $V$ is a radial potential and $E(t)$ a circularly polarized field with uniform frequency $ω$. The quantum monodromy operator (QMO) that takes the system through a complete period $T=2π/ω$, turns out to be unitarily equivalent to $e^{iTP_A(x,hD_x)/h}$, where $P_A(x,hD_x))$ identifies with a magnetic Schrödinger operator. When $V$ is sufficiently confining, $P_A(x,hD_x))$ presents a double magnetic well. Then we construct its semi-classical ground state and examine the splitting between its two first eigenvalues.

math-ph↗

arXiv2019

Semi-classical quantum maps of semi-hyperbolic type

Let M = R n or possibly a Riemannian, non compact manifold. We consider semi-excited resonances for a h-differential operator H(x, hD x ; h) on L 2 (M) induced by a non-degenerate periodic orbit $γ$ 0 of semi-hyperbolic type, which is contained in the non critical energy surface {H 0 = 0}. By semi-hyperbolic, we mean that the linearized Poincar{é} map dP 0 associated with $γ$ 0 has at least one eigenvalue of modulus greater (or less) than 1, and one eigenvalue of modulus equal to 1, and by non-degenerate that 1 is not an eigenvalue, which implies a family $γ$(E) with the same properties. It is known that an infinite number of periodic orbits generally cluster near $γ$ 0 , with periods approximately multiples of its primitive period. We construct the monodromy and Grushin operator, adapting some arguments by [NoSjZw], [SjZw], and compare with those obtained in [LouRo], which ignore the additional orbits near $γ$ 0 , but still give the right quantization rule for the family $γ$(E).

math.AP↗

arXiv2018

Bohr-Sommerfeld quantization rules revisited: the method of positive commutators

We revisit the well known Bohr-Sommerfeld quantization rule (BS) for a 1-D Pseudo-differential self-adjoint Hamiltonian within the algebraic and microlocal framework of Helffer and Sjöstrand; BS holds precisely when the Gram matrix consisting of scalar products of some WKB solutions with respect to the "flux norm" is not invertible. The interest of this procedure lies in its possible generalization to matrix-valued Hamiltonians, like Bogoliubov-de Gennes Hamiltonian. It is simplified in the scalar case by using action-angle variables.

math-ph↗

arXiv2016

Semi-classical quantization rules for a periodic orbit of hyperbolic type

Determination of periodic orbits for a Hamiltonian system together with their semi-classical quantization has been a long standing problem. We consider here resonances for a $h$-Pseudo-Differential Operator $H(y,hD_y;h)$ induced by a periodic orbit of hyperbolic type at energy $E_0$. We generalize the framework of [GéSj], in the sense that we allow for both hyperbolic and elliptic eigenvalues of Poincaré map, and show that all resonances in $W=[E_0-\varepsilon_0,E_0+\varepsilon_0]-i]0,h^δ]$, $0<δ<1$, are given by a generalized Bohr-Sommerfeld quantization rule.

math-ph↗

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↗