Algebraic spectral gaps

For the one-dimensional Schr\"odinger equation, some real intervals with no eigenvalues (the spectral gaps) may be obtained rather systematically with a method proposed by H. Giacomini and A. Mouchet in 2007. The present article provides some alternative formulation of this method, suggests some possible generalizations and extensively discusses the higher-dimensional case.Comment: Submitted to ESAIM PROCEEDING

Upper and lower bounds for an eigenvalue associated with a positive eigenvector

When an eigenvector of a semi-bounded operator is positive, we show that a remarkably simple argument allows to obtain upper and lower bounds for its associated eigenvalue. This theorem is a substantial generalization of Barta-like inequalities and can be applied to non-necessarily purely quadratic Hamiltonians. An application for a magnetic Hamiltonian is given and the case of a discrete Schrodinger operator is also discussed. It is shown how this approach leads to some explicit bounds on the ground-state energy of a system made of an arbitrary number of attractive Coulombian particles

A mechanical model of tunnelling

It is shown how the model which was introduced by Mouchet (2008 Eur. J. Phys. 29 1033) allows one to mimic the quantum tunnelling between two symmetric one-dimensional wells

Applications of Noether conservation theorem to Hamiltonian systems

The Noether theorem connecting symmetries and conservation laws can be applied directly in a Hamiltonian framework without using any intermediate Lagrangian formulation. This requires a careful discussion about the invariance of the boundary conditions under a canonical transformation and this paper proposes to address this issue. Then, the unified treatment of Hamiltonian systems offered by Noether's approach is illustrated on several examples, including classical field theory and quantum dynamics.Comment: Version

Importance of the Wick rotation on Tunnelling

A continuous complex rotation of time t\mapsto t\EXP{-i\theta} is shown to smooth out the huge fluctuations that characterise chaotic tunnelling. This is illustrated in the kicked rotor model (quantum standard map) where the period of the map is complexified: the associated chaotic classical dynamics, if significant for $\theta=0$, is blurred out long before the Wick rotation is completed ($\theta=\pi/2$). The influence of resonances on tunnelling rates weakens exponentially as $\theta$ increases from zero, all the more rapidly the sharper the fluctuations. The long range fluctuations can therefore be identified in a deterministic way without ambiguity. When the last ones have been washed out, tunnelling recovers the (quasi-)integrable exponential behaviour governed by the action of a regular instanton.Comment: 4 figure

Variations on chaos in physics: from unpredictability to universal laws

The tremendous popular success of Chaos Theory shares some common points with the not less fortunate Relativity: they both rely on a misunderstanding. Indeed, ironically , the scientific meaning of these terms for mathematicians and physicists is quite opposite to the one most people have in mind and are attracted by. One may suspect that part of the psychological roots of this seductive appeal relies in the fact that with these ambiguous names, together with some superficial clich{\'e}s or slogans immediately related to them ("the butterfly effect" or "everything is relative"), some have the more or less secret hope to find matter that would undermine two pillars of science, namely its ability to predict and to bring out a universal objectivity. Here I propose to focus on Chaos Theory and illustrate on several examples how, very much like Relativity, it strengthens the position it seems to contend with at first sight: the failure of predictability can be overcome and leads to precise, stable and even more universal predictions.Comment: Convegno "Matematica e Cultura 2015", Mar 2015, Venezia, Ital

Finding gaps in a spectrum

We propose a method for finding gaps in the spectrum of a differential operator. When applied to the one-dimensional Hamiltonian of the quartic oscillator, a simple algebraic algorithm is proposed that, step by step, separates with a remarkable precision all the energies even for a double-well configuration in a tunnelling regime. Our strategy may be refined and generalised to a large class of 1d-problems

An alternative proof of Wigner theorem on quantum transformations based on elementary complex analysis

International audienceAccording to Wigner theorem, transformations of quantum states which preserve the probabilities are either unitary or antiunitary. This short communication presents an elementary proof of this theorem that significantly departs from the numerous ones already existing in the literature. The main line of the argument remains valid even in quantum field theory where Hilbert spaces are non-separable

Introduction aux méthodes semiclassiques en chaos quantique

On replace les méthodes semiclassiques en chaos quantique dans une perspective historique, peu technique, étayée par une bibliographie abondante mais non exhaustive allant jusqu'au milieu des années 90