Non linear eigenvalue problems

In this paper we consider generalized eigenvalue problems for a family of operators with a polynomial dependence on a complex parameter. This problem is equivalent to a genuine non self-adjoint operator. We discuss here existence of non trivial eigenstates for models coming from analytic theory of smoothness for P.D.E. We shall review some old results and present recent improvements on this subject

Time Evolution of States for Open Quantum Systems. The quadratic case

Our main goal in this paper is to extend to any system of coupled quadratic Hamiltonians some properties known for systems of quantum harmonic oscillators related with the Brownian Quantum Motion model. In a first part we get a rather general formula for the purity (or the linear entropy) in a short time approximation. In a second part we establish a master equation (or a Fokker-Planck type equation) for the time evolution of the reduced matrix density for bilinearly coupled quadratic Hamiltonians. The Hamiltonians and the bilinear coupling can be time dependent. Moreover we give an explicit formula for the solution of this master equation so that the time evolution of the reduced density at time $t$ is connected with the reduced density at initial time $t_0$ for $t_0 \leq t <t_0 +t_c$ where $t_c\in ]0, \infty]$ is a critical time but reversibility is lost for $t \geq t_0 +t_c$

Irregular time dependent perturbations of quantum Hamiltonians

Our main goal in this paper is to prove existence (and uniqueness) of the quantum propagator for time dependent quantum Hamiltonians $\hat H(t)$ when this Hamiltonian is perturbed with a quadratic white noise $\dot{\beta}\hat K$. $\beta$ is a continuous function in time $t$, $\dot \beta$ its time derivative and $K$ is a quadratic Hamiltonian. $\hat K$ is the Weyl quantization of $K$. For time dependent quadratic Hamiltonians $H(t)$ we recover, under less restrictive assumptions, the results obtained in \cite{bofu, du}.In our approach we use an exact Hermann Kluk formula \cite{ro2} to deduce a Strichartz estimate for the propagator of $\hat H(t) +\dot \beta K$. This is applied to obtain local and global well posedness for solutions for non linear Schr\"odinger equations with an irregular time dependent linear part

Random weighted Sobolev inequalities and application to quantum ergodicity

This paper is a continuation of Poiret-Robert-Thomann (2013) where we studied a randomisation method based on the Laplacian with harmonic potential. Here we extend our previous results to the case of any polynomial and confining potential $V$ on $\mathbb{R}^d$. We construct measures, under concentration type assumptions, on the support of which we prove optimal weighted Sobolev estimates on $\mathbb{R}^d$. This construction relies on accurate estimates on the spectral function in a non-compact configuration space. Then we prove random quantum ergodicity results without specific assumption on the classical dynamics. Finally, we prove that almost all basis of Hermite functions is quantum uniquely ergodic.Comment: Clarifications added in the part concerning QU

Quadratic Quantum Hamiltonians revisited

Time dependent quadratic Hamiltonians are well known as well in classical mechanics and in quantum mechanics. In particular for them the correspondance between classical and quantum mechanics is exact. But explicit formulas are non trivial (like the Mehler formula). Moreover, a good knowlege of quadratic Hamiltonians is very useful in the study of more general quantum Hamiltonians and associated Schr\"{o}dinger equations in the semiclassical regime. Our goal here is to give our own presentation of this important subject. We put emphasis on computations with Gaussian coherent states. Our main motivation to do that is application concerning revivals and Loschmidt echo

Leapfrogging vortex rings for the three dimensional Gross-Pitaevskii equation

Leapfrogging motion of vortex rings sharing the same axis of symmetry was first predicted by Helmholtz in his famous work on the Euler equation for incompressible fluids. Its justification in that framework remains an open question to date. In this paper, we rigorously derive the corresponding leapfrogging motion for the axially symmetric three-dimensional Gross-Pitaevskii equation.Comment: 39 pages, 2 figure

Supersymmetry and Ghosts in Quantum Mechanics

2000 Mathematics Subject Classification: 81Q60, 35Q40.A standard supersymmetric quantum system is defined by a Hamiltonian [^H] = ½([^Q]*[^Q] +[^Q][^Q]*), where the super-charge [^Q] satisfies [^Q]2 = 0, [^Q] commutes with [^H]. So we have [^H] ≥ 0 and the quantum spectrum of [^H] is non negative. On the other hand Pais-Ulhenbeck proposed in 1950 a model in quantum-field theory where the d'Alembert operator [¯] = [(∂2)/( ∂t2)] − Δx is replaced by fourth order operator [¯]([¯] + m2), in order to eliminate the divergences occuring in quantum field theory. But then the Hamiltonian of the system, obtained by second quantization, has large negative energies called "ghosts" by physicists. We report here on a joint work with A. Smilga (SUBATECH, Nantes) where we consider a similar problem for some models in quantum mechanics which are invariant under supersymmetric transformations. We show in particular that "ghosts" are still present