Séminaire de physique mathématique (archives)

Nous rapportons sur des résultats de non-unicité, d’unicité et de reconstruction pour le problème de diffusion inverse sans information de phase. Nous sommes motivés par un progrès récent et très essentiel dans ce domaine.

Abstract：In this talk we consider the spectral property of a Fokker-Planck operator with potential. By virtue of a multiplier method inspired by Nicolas Lerner, we obtain new compactness criteria for its resolvent, involving the control of the positive eigenvalues of the Hessian matrix of the potential.

Given a quantum particle on a line, its momentum and position are described by a pair of Hermitean operators (p, q) which satisfy the canonical commuta-tion relation. There is a third observable r, say, contained in the Heisenberg algebra generated by p and q, which simultaneously satisfies canonical com-mutation relations with both position and momentum. The Heisenberg triple of the observables (p, q, r) is not only unique (up to unitary equivalences) but also maximal (no four equi-commutant observables exist). Being invariant under a cyclic permutation, the triple (p, q, r) endows the Heisenberg algebra with an interesting threefold, largely unexplored symmetry. I will briefly sketch why these considerations are important in the context of so-called mutually unbiased bases, and that they suggest to rethink Heisenberg's uncertainty relation by first generalizing it to an expression involving the pro-duct of three variances, and then to even more general functions thereof.

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We exhibit elementary facts about 4D supersymmetric theories with emphasis on N = 1 supersymmetry and argue that the adequate language to deal with this type of theories is the superspace approach. Contents: From symmetries to super-symmetry; Basic features of supersymmetry; Superspace and superelds; 1D supersymmetry as the simplest example.

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The early universe cosmology can be successfully described in the theoretical framework of modified gravity and quintessence. I introduce the Starobinsky and Linde inflationary models in light of the recent CMB observations by the PLANCK satellite mission and the BICEP2 telescope. Preheating and reheating after inflation are briefly reviewed. Some very recent theoretical results about inflation, leptogenesis, dark matter and dark energy in the context of N=1 supergravity are outlined.

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One of the key tenets of standard quantum mechanics is Hermiticity, which, among other things, guarantees the reality of energy eigenvalues. However, there exists a whole class of Hamiltonians which are not Hermitian but nonetheless possess a completely real spectrum. These Hamiltonians, of which the paradigm is $H=p^2+ix^3$, are PT symmetric, whereby $x$ goes to $-x$ and $i$ to $-i$. I will review the status of such Hamiltonians, which have been the subject of intensive study over the last few years. An unexpected development was the realization that ideas developed in the context of quantum mechanics could be applied to classical optics. There is a standard approximation in optics, the paraxial approximation, where the equation for propagation has the form of an analogue Schroedinger equation, with the longitudinal distance $z$ playing the role of time and the refractive index taking the role of the potential. PT symmetry implies a medium with both gain and loss balanced in a particular way. The advantage is that real eigenvalues correspond to propagation without exponential growth or decay. Artificial PT-symmetric media have many unusual and potentially useful properties.

In a celebrated 1961 paper, Landauer formulated a fundamental lower bound on the energy dissipated by computation processes. Since then, there has been many attempts to formalize, generalize and contradict Landauer's analysis. The situation became even worse with the advent of quantum computing. In a recent enlightening article, Reeb and Wolf sets the game into the framework of quantum statistical mechanics, and finally gave a precise mathematical formulation of Landauer's bound. I will discuss parts of this analysis and present some extensions of it that were obtained in a joint work with V. Jaksic.