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André Martinez

Publications and source records attributed to André Martinez.

15 recordsLinked to original sources
arXiv2019

Widths of resonances above an energy-level crossing

We study the existence and location of the resonances of a $2\times 2$ semiclassical system of coupled Schrödinger operators, in the case where the two electronic levels cross at some point, and one of them is bonding, while the other one is anti-bonding. Considering energy levels just above that of the crossing, we find the asymptotics of both the real parts and the imaginary parts of the resonances close to such energies. This is a continuation of our previous works where we considered energy levels around that of the crossing.

math-ph↗
arXiv2018

Molecular dynamics at an energy-level crossing

This paper is a continuation of a previous work about the study of the survival probability modelizing the molecular predissociation in the Born-Oppenheimer framework. Here we consider the critical case where the reference energy corresponds to the value of a crossing of two electronic levels, one of these two levels being confining while the second dissociates. We show that the survival probability associated to a certain initial state is a sum of the usual time-dependent exponential contribution, and a reminder term that is jointly polynomially small with respect to the time and the semiclassical parameter. We also compute explicitly the main contribution of the remainder.

math-ph↗
arXiv2018

Widths of highly excited resonances in multidimensional molecular predissociation

We investigate the simple resonances of a 2 by 2 matrix of n-dimensional semiclassical Shrödinger operators that interact through a first order differential operator. We assume that one of the two (analytic) potentials admits a well with non empty interior, while the other one is non trapping and creates a barrier between the well and infinity. Under a condition on the resonant state inside the well, we find an optimal lower bound on the width of the resonance. The method of proof relies on Carleman estimates, microlocal propagation of the microsupport, and a refined study of a non involutive double characteristic problem in the framework of Sjöstrand's analytic microlocal theory.

math.AP↗
arXiv2017

Excited resonance widths for Helmholtz resonators with straight neck

We consider resonances associated with excited eigenvalues of the cavity of a general Helmholtz resonator with straight neck. Under the assumption that the neck stays away from the nodal set of the corresponding eigenstate, we generalise the optimal exponential lower bound on the width of the resonance, that we have obtained in a previous paper for the ground resonance only.

math.AP↗
arXiv2016

Widths of highly excited shape resonances

We study the widths of shape resonances for the semiclassical multi-dimensional Schrödinger operator, in the case where the frequency remains close to some value strictly larger than the bottom of the well. Under a condition on the behavior of the resonant state inside the well, we obtain an optimal lower bound for the widths.

math-ph↗
arXiv2015

Estimates on the molecular dynamics for the predissociation process

We study the survival probability associated with a semi-classical matrix Shrödinger operator that models the predissociation of a general molecule in the Born-Oppenheimer approximation. We show that it is given by its usual time-dependent exponential contribution, up to a reminder term that is exponentially small (in the semiclassical parameter) with arbitrarily large rate of decay. The result applies in any dimension, and in presence of a number of resonances that may tend to infinity as the semiclassical parameter tends to 0.

math.SP↗
arXiv2015

Molecular predissociation resonances at an energy-level crossing

We study the resonances of a two-by-two semiclassical system of one dimensional Schrödinger operators, near an energy where the two potentials intersect transversally, one of them being bonding, and the other one anti-bonding. Under an ellipticity condition on the interaction, we obtain optimal estimates on the location and on the widths of these resonances.

math-ph↗
arXiv2015

On the Born-Oppenheimer approximation of diatomic molecular resonances

We give a new reduction of a general diatomic molecular Hamiltonian, without modifying it near the collision set of nuclei. The resulting effective Hamiltonian is the sum of a smooth semiclassical pseudodifferential operator (the semiclassical parameter being the inverse of the square-root of the nuclear mass), and a semibounded operator localised in the elliptic region corresponding to the nuclear collision set. We also study its behaviour on exponential weights, and give several applications where molecular resonances appear and can be well located.

math-ph↗
arXiv2015

R&D activities on RF contacts for the ITER ion cyclotron resonance heating launcher

Embedded RF contacts are integrated within the ITER ICRH launcher to allow assembling, sliding and to lower the thermo-mechanical stress. They have to withstand a peak RF current up to 2.5 kA at 55 MHz in steady-state conditions, in the vacuum environment of themachine.The contacts have to sustain a temperature up to 250{\textdegree}Cduring several days in baking operations and have to be reliable during the whole life of the launcher without degradation. The RF contacts are critical components for the launcher performance and intensive R&D is therefore required, since no RF contactshave so far been qualified at these specifications. In order to test and validate the anticipated RF contacts in operational conditions, CEA has prepared a test platform consisting of a steady-state vacuum pumped RF resonator. In collaboration with ITER Organization and the CYCLE consortium (CYclotronCLuster for Europe), an R&D program has been conducted to develop RF contacts that meet the ITER ICRH launcher specifications. A design proposed by CYCLE consortium, using brazed lamellas supported by a spring to improve thermal exchange efficiency while guaranteeing high contact force, was tested successfully in the T-resonator up to 1.7 kA during 1200 s, but failed for larger current values due to a degradation of the contacts. Details concerning the manufacturing of the brazed contacts on its titanium holder, the RF tests results performed on the resonator and the non-destructive tests analysis of the contacts are given in this paper.

physics.ins-det↗
arXiv2012

Resonance widths in a case of multidimensional phase space tunneling

We consider a semiclassical $2\times 2$ matrix Schrödinger operator of the form $P=-h^2Δ{\bf I}_2 + {\rm diag}(x_n-μ, τV_2(x)) +hR(x,hD_x)$, where $μ$ and $τ$ are two small positive constants, $V_2$ is real-analytic and admits a non degenerate minimum at 0, and $R=(r_{j,k}(x,hD_x))_{1\leq j,k\leq 2}$ is a symmetric off-diagonal $2\times 2$ matrix of first-order differential operators with analytic coefficients. Then, denoting by $e_1$ the first eigenvalue of $-Δ+ \la τV_2"(0)x,x\ra /2$, and under some ellipticity condition on $r_{1,2}=r_{2,1}^*$, we show that, for any $μ$ sufficiently small, and for $0<τ\leqτ(μ)$ with some $τ(μ)>0$, the unique resonance $ρ$ of $P$ such that $ρ= τV_2(0) + (e_1+r_{2,2}(0,0))h + {\mathcal O}(h^2)$ (as $h\rightarrow 0_+$) satisfies, $$ \Im ρ= -h^{\frac32}f(h,\ln\frac1{h})e^{-2S/h}, $$ where $f(h,\ln\frac1{h}) \sim \sum_{0\leq m\leq\ell} f_{\ell,m}h^\ell(\ln\frac1{h})^m$ is a symbol with $f_{0,0}>0$, and $S$ is the imaginary part of the complex action along some convenient closed path containing $(0,0)$ and consisting of a union of complex nul-bicharacteristics of $p_1:=ξ^2 - x_n-μ$ and $p_2:=ξ^2 +τV_2(x)$ (broken instanton). This broken instanton is described in terms of the outgoing and incoming complex Lagrangian manifolds associated with $p_2$ at the point $(0,0)$, and their intersections with the characteristic set $p_1^{-1}(0)$ of $p_1$.

math-ph↗
arXiv2012

Resonance widths for the molecular predissociation

We consider a semiclassical $2\times 2$ matrix Schrödinger operator of the form $P=-h^2Δ{\bf I}_2 + {\rm diag}(V_1(x), V_2(x)) +hR(x,hD_x)$, where $V_1, V_2$ are real-analytic, $V_2$ admits a non degenerate minimum at 0, $V_1$ is non trapping at energy $V_2(0)=0$, and $R(x,hD_x)=(r_{j,k}(x,hD_x))_{1\leq j,k\leq 2}$ is a symmetric off-diagonal $2\times 2$ matrix of first-order pseudodifferential operators with analytic symbols. We also assume that $V_1(0) >0$. Then, denoting by $e_1$ the first eigenvalue of $-Δ+ \la V_2"(0)x,x\ra /2$, and under some ellipticity condition on $r_{1,2}$ and additional generic geometric assumptions, we show that the unique resonance $ρ_1$ of $P$ such that $ρ_1 = V_2(0) + (e_1+r_{2,2}(0,0))h + {\mathcal O}(h^2)$ (as $h\rightarrow 0_+$) satisfies, $$ \Im ρ_1 = -h^{n_0+(1-n_Γ)/2}f(h,\ln\frac1{h})e^{-2S/h}, $$ where $f(h,\ln\frac1{h}) \sim \sum_{0\leq m\leq\ell} f_{\ell,m}h^\ell(\ln\frac1{h})^m$ is a symbol with $f_{0,0}>0$, $S>0$ is the so-called Agmon distance associated with the degenerate metric $\max(0, \min(V_1,V_2))dx^2$, between 0 and $\{V_1\leq 0\}$, and $n_0\geq 1$, $n_Γ\geq 0$ are integers that depend on the geometry.

math-ph↗
arXiv2012

The spectrum of the cubic oscillator

We prove the simplicity and analyticity of the eigenvalues of the cubic oscillator Hamiltonian,$H(β)=-d^2/dx^2+x^2+i\sqrtβx^3$,for $β$ in the cut plane $\C_c:=\C\backslash (-\infty, 0)$. Moreover, we prove that the spectrum consists of the perturbative eigenvalues $\{E_n(β)\}_{n\geq 0}$ labeled by the constant number $n$ of nodes of the corresponding eigenfunctions. In addition, for all $β\in\C_c$, $E_n(β)$ can be computed as the Stieltjes-Padé sum of its perturbation series at $β=0$. This also gives an alternative proof of the fact that the spectrum of $H(β)$ is real when $β$ is a positive number. This way, the main results on the repulsive PT-symmetric and on the attractive quartic oscillators are extended to the cubic case.

math-ph↗
arXiv2008

Resonances for non-analytic potentials

We consider semiclassical Schroedinger operators on R^n, with C^\infty potentials decaying polynomially at infinity. The usual theories of resonances do not apply in such a non-analytic framework. Here, under some additional conditions, we show that resonances are invariantly defined up to any power of their imaginary part. The theory is based on resolvent estimates for families of approximating distorted operators with potentials that are holomorphic in narrow complex sectors around R^n.

math.SP↗