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Michel Rouleux

Publications and source records attributed to Michel Rouleux.

At least 19 recordsLinked to original sources
arXiv2026

Asymptotic solutions: glancing trajectories, Lagrangian singularities, and Bessel cylinders

We find a normal form for the simplest Lagrangian singularity appearing as the projection onto the space-energy of a solution to the Hamilton-Jacobi equation. Using this normal form we approximate asymptotic solutions of an equation $(\widehat{H}-E_0) \, u_h = f_h$ with a semiclassical distribution $f_h$ microlocalized on a Lagrangian manifold $Λ_0$ when $E_0$ is a critical value of the restriction $H|_{Λ_0}$.

math-ph↗
arXiv2026

The flux norm, Bohr-Sommerfeld Quantization Rules and the scattering problem for h $Ψ$DO's on the real line

We revisit the well known Bohr-Sommerfeld quantization rule (BS) of order 2 for a self-adjoint 1-D h-Pseudo-differential operator within the algebraic and microlocal framework of Helffer and Sjoestrand; BS holds precisely when Gram matrix consisting of scalar products of some WKB solutions with respect to the ``flux norm'' (or microlocal Wronskian) is not invertible. We simplify somewhat our previous proof [A. Ifa H. Louati and M. Rouleux. Bohr-Sommerfeld Quantization Rules Revisited: the Method of Positive Commutators. J. Math. Sci. Univ. Tokyo, 25(2):2018] by working in spatial representation only, as in complex WKB theory for Schroedinger operator. We consider also the scattering problem.

math-ph↗
arXiv2025

Semiclassical Green functions and Lagrangian intersection. Applications to the propagation of Bessel beams in non-homogeneous media

We study semi-classical asymptotics for problems with localized right-hand sides by considering a Hamiltonian $H(x,p)$ positively homogeneous of degree $m\geq1$ on $T^*{\bf R}^n\setminus0$. The energy shell is $H(x,p)=E$, and the right-hand side $f_h$ is microlocalized: (1) on the vertical plane $Λ_0=\{x=x_0\}$; (2) on the ``cylinder'' $Λ_0=\{(X,P)=\bigl(φω(ψ),ω(ψ)\bigr); \ φ\in {\bf R}, ω(ψ)=(\cosψ,\sinψ)\}$. when $n=2$. Most precise results are obtained in the isotropic case $H(x,p)={|p|^m\overρ(x)}$, with $ρ$ a smooth positive function. In case (2), $Λ_0$ is the frequency set of Bessel function $J_0({|x|\over h})$, and the solution $u_h$ of $(H(x,hD_x)-E)u_h=f_h$ when $m=1$, already provides an insight in the structure of ``Bessel beams'', which arise in the theory of optical fibers. We present in this work some extensions of A.Anikin, S.Dobrokhotov, V.Nazaikinskii, M.Rouleux, Theor. Math. Phys. 214(1): p.1-23, 2023. In Sect.3 we sketch the semi-classical counterpart of the construction of parametrices for the Cauchy problem with Lagrangian intersections, as is set up by R.Melrose and G.Uhlmann. This involves Maslov {\it bi-canonical operator}.

math.AP↗
arXiv2024

Generalized exchange operators for a system of spin-1 particles

The irreps $(SU(2),{\cal H},U)$ of SU(2) of dimension $(2S+1)^N$, i.e. operators acting on the space ${\cal H}={\cal H}_N={\bf C}^{(2S+1)^N}$ of $N$ identical particles with spin $S$, are described by Clebsch-Gordan decomposition into inequivalent irreps. In the special case $S=1/2$, Dirac \cite{Dir1} discovered that there is another rep given by $({\cal S}(N),{\cal H},V)$ where ${\cal S}(N)$ is the permutation group, Thus, the standard ``linear'' Hamiltonian, or Heisenberg interaction Hamiltonian $H_0=\sum_{1\leq i\leq N}\vec S_i\cdot\vec S_j$, where $\vec σ_i=2\vec S_i$ is the vector of Pauli matrices, can be interpreted as the sum of the ``Exchange Operators'' $P_{ij}$ between particles $i$ and $j$. Schrödinger \cite{Sch} generalized to higher spin numbers $S$ the Exchange Operator $P_{ij}=P_S(\vec S_i\cdot \vec S_j)$ as a polynomial of degree $2S$ in $\vec S_i\cdot \vec S_j$. This we call the $P$-representation. There is another rep induced by the one particle permutation of states operators $\widetilde Q_α$, which we call the $Q$-rep. Our main purpose is to write some physical Hamiltonians for a few particles in the $P$- or $Q$-rep and compute their spectrum. The simplest case where there are as many particles as available states for the spin operator along the $z$-axis, i.e. $N=2S+1=3$, see Weyl \cite{Wey} or Hamermesh \cite{Ham}. Finally, we consider the relationship between permutations and rotation invariance when $S=1/2$ and $S=1$.

math-ph↗
arXiv2023

Tunneling for a semi-classical magnetic Schrödinger operator with symmetries

We are interested in decay estimates of the ground state (or the low energy eigenstates), outside the potential wells, for a semi-classical Magnetic Schrödinger operator with smooth coefficients $P_A(x,hD_x)=(hD_x-μA(x))^2+V(x)$ on $L^2({\bf R}^d)$. We shall essentially consider the case where $μ$ is large. This kind of estimates, in case of Schrödinger operator without a magnetic field, have been studied by Agmon, also in the case of a Riemannian manifold $M$. Agmon estimates hold true for any $h$, but are particularly useful in the limit $h\to0$ when studying tunneling.

math-ph↗
arXiv2022

Generalized Pell-Fermat equations and Pascal triangle

Using Pascal triangle, we give a simple generalization to the so-called STRAND Puzzle solved by Srinivasa Ramanujan. Thus we are interested in computing the median, first and third quartiles of some integer valued distributions, arising naturally when extending partial sums of the arithmetic progression (triangular numbers) to tetrahedral numbers and beyond. We show this reduces to equations of Pell-Fermat type of higher order, which admit very few integer solutions, but for which, following Ramanujan's original idea, we can always find integer sequences of best approximation, in the Diophantine sense. In absence of a general theory on Pell-Fermat equation of higher order, our procedure relies much on formal Calculus with Mathematica.

math.NT↗
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

Multimode entanglement for fermions

We are motivated by tripartite entanglement for fermions. While GHZ or W states involve 3-fold intrication, we consider here piecewise intrication of 3 fermions in ${\bf C}^2$, namely of type $ab+bc+ca$. Before interaction with Stern-Gerlach apparatus, qu-bits are distinguishable; at the output however they turn into un-distinguishable particles, whose anti-symmetric wave function is of the form $\det(b-a,c-a)$ (affine determinant). More generally, $d+1$ intricated fermions in ${\bf C}^d$ can be represented by the anti-symmetric wave function $\det(a_1-a_0,a_2-a_0,\cdots,a_d-a_0)$. We investigate also properties of affine Slater determinants, as expectation values or reduced density matrices.

quant-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

The one dimensional semi-classical Bogoliubov-de Gennes Hamiltonian with PT symmetry: generalized Bohr-Sommerfeld quantization rules

We present a method for computing first order asymptotics of semiclassical spectra for 1-D Bogoliubov-de Gennes (BdG) Hamiltonian from Supraconductivity, which models the electron/hole scattering through two SNS junctions. This involves: 1) reducing the system to Weber equation near the branching point at the junctions, 2) constructing local sections of the fibre bundle of microlocal solutions, 3) normalizing these solutions for the "flux norm" associated to the microlocal Wronskians, 4) finding the relative monodromy matrices in the gauge group that leaves invariant the flux norm, 5) from this we deduce Bohr-Sommerfeld (BS) quantization rules that hold precisely when the fibre bundle of microlocal solutions (depending on the energy parameter E) has trivial holonomy. Such a semi-classical treatement reveals interesting continuous symetries related to monodromy.

math-ph↗
arXiv2018

Semi-classical Green functions

Let $H(x,p)\sim H_0(x,p)+hH_1(x,p)+\cdots$ be a semi-classical Hamiltonian on $T^*{\bf R}^n$, and $Σ_E=\{H_0(x,p)=E\}$ a non critical energy surface. Consider $f_h$ a semi-classical distribution (the "source") microlocalized on a Lagrangian manifold $Λ$ which intersects cleanly the flow-out $Λ_+$ of the Hamilton vector field $X_{H_0}$ in $Σ_E$. Using Maslov canonical operator, we look for a semi-classical distribution $u_h$ satisfying the limiting absorption principle and $H^w(x,hD_x)u_h=f_h$ (semi-classical Green kernel). In this report, we elaborate (still at an early stage) on some results announced in [Doklady Akad. Nauk, Vol. 76, No1, p.1-5, 2017] and provide some examples, in particular from the theory of wave beams.

math-ph↗
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↗
arXiv2017

Generalized Pareto optimum and semi-classical spinors

In 1971, S.Smale presented a generalization of Pareto optimum he called the critical Pareto set. The underlying motivation was to extend Morse theory to several functions, i.e. to find a Morse theory for $m$ differentiable functions defined on a manifold $M$ of dimension $\ell$. We use this framework to take a $2\times2$ Hamiltonian ${\cal H}={\cal H}(p)\in C^\infty(T^*{\bf R}^2)$ to its normal form near a singular point of the Fresnel surface. Namely we say that ${\cal H}$ has the Pareto property if it decomposes, locally, up to a conjugation with regular matrices, as ${\cal H}(p)=u'(p)C(p)(u'(p))^*$, where $u:{\bf R}^2\to{\bf R}^2$ has singularities of codimension 1 or 2, and $C(p)$ is a regular Hermitian matrix ("integrating factor"). In particular this applies in certain cases to the matrix Hamiltonian of Elasticity theory and its (relative) perturbations of order 3 in momentum at the origin.

math-ph↗
arXiv2017

Asymptotics of Green function for the linear waves equations in a domain with a non-uniform bottom

We consider the linear problem for water-waves created by sources on the bottom and the free surface in a 3-D basin having slowly varying profile $z=-D(x)$. The fluid verifies Euler-Poisson equations. These (non-linear) equations have been given a Hamiltonian form by Zakharov, involving canonical variables $(ξ(x,t),η(x,t))$ describing the dynamics of the free surface; variables $(ξ,η)$ are related by the free surface Dirichlet-to-Neumann (DtN) operator. For a single variable $x\in{\bf R}$ and constant depth, DtN operator was explicitely computed in terms of a convergent series. Here we neglect quadratic terms in Zakharov equations, and consider the linear response to a disturbance of $D(x)$ harmonic in time when the wave-lenght is small compared to the depth of the basin. We solve the Green function problem for a matrix-valued DtN operator, at the bottom and the free-surface.

math-ph↗
arXiv2016

Quantum Vorticity at positive temperature for spin systems with continuous symmetry

We propose a definition of vorticity at inverse temperature $β$ for Gibbs states in quantum XY or Heisenberg spin systems on the lattice by testing $\exp[-βH]$ on a complete set of observables ("one-point functions"). Imposing a compression of Pauli matrices at the boudary, which stands for the classical environment, we perform some numerical simulations on finite lattices in case of XY model, which exhibit usual vortex patterns.

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↗