Endomorphisme autoadjoint

Résumé

En mathématiques et plus précisément en algèbre linéaire, un endomorphisme autoadjoint ou opérateur hermitien est un endomorphisme d'espace de Hilbert qui est son propre adjoint (sur un espace de Hilbert réel on dit aussi endomorphisme symétrique). Le prototype d'espace de Hilbert est un espace euclidien, c'est-à-dire un espace vectoriel sur le corps des réels, de dimension finie, et muni d'un produit scalaire. L'analogue sur le corps des complexes s'appelle un espace hermitien. Sur ces espaces de Hilbert de dimension finie, un endomorphisme autoadjoint est diagonalisable dans une certaine base orthonormale et ses valeurs propres (même dans le cas complexe) sont réelles. Les applications des propriétés structurelles d'un endomorphisme autoadjoint (donc de sa forme quadratique associée) sont nombreuses. Définition Espace de Hilbert Une telle application admet donc un adjoint a^* (égal à a,), si bien que c'est un endomorphisme de H : elle est autom

Source officielle

https://fr.wikipedia.org/wiki/Endomorphisme_autoadjoint

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Aspects of two dimensional magnetic Schrödinger operatorsquantum hall systems and magnetic Stark resonances

Christian Ferrari

In this PhD thesis we deal with two mathematical problems arising from quantum mechanics. We consider a spinless non relativistic quantum particle whose configuration space is a two dimensional surface S. We also suppose that the particle feels the effect of an homogeneous magnetic field perpendicular to the surface S. In the first case S = R × SL1, the infinite cylinder of circumference L, corresponding to periodic boundary conditions, while in the second one S = R2. In both cases the particle feels the effect of an additional suitable potential. We are thus left with the study of two specific classes of Schrödinger operators. The operator of the first class generates the dynamics of the particle when it is submitted to an Anderson-type random potential, as well as to a non random potential confining the particle along the cylinder axis in an interval of length L. In this case we describe the spectrum and classify it by the quantum mechanical current carried by the corresponding eigenfunctions. We prove that there are spectral regions in which all the eigenvalues have an order one current with respect to L, and spectral regions where eigenvalues with order one current and eigenvalues with infinitesimal current with respect to L are intermixed. These results are relevant for the theory of the integer quantum Hall effect. The second Schrödinger operator class corresponds to the physical situation where the potential is the sum of a "local" potential and of a potential due to a weak constant electric field F. In this case we show that the resonant states, induced by the electric field, decay exponentially at a rate given by the imaginary part of the eigenvalues of some non self-adjoint operator. Moreover we prove an upper bound on this imaginary part that turns out to be of order exp(-1/F2) as F goes to zero. Therefore the lifetime of the resonant states is at least of order exp(-1/F2).

EPFL2003

Chargement

In this note we provide an alternative way of defining the self-adjoint Hamiltonian of the harmonic oscillator perturbed by an attractive delta'-interaction, of strength beta, centred at 0 (the bottom of the confining parabolic potential), that was rigorously defined in a previous paper by means of a 'coupling constant renormalisation'. Here we get the Hamiltonian as a norm resolvent limit of the harmonic oscillator Hamiltonian perturbed by a triple of attractive delta-interactions, thus extending the Cheon-Shigehara approximation to the case in which a confining harmonic potential is present.

Iop Publishing Ltd2016

Chargement

We perform a detailed study of double Higgs production via gluon fusion in the effective field theory (EFT) framework where effects from new physics (NP) are parametrized by local operators. Our analysis provides a perspective broader than the one followed in most of the previous analyses, where this process was merely considered as a way to extract the Higgs trilinear coupling. We focus on the hh -> b (b) over bar gamma gamma channel and perform a thorough simulation of signal and background at the 14 TeV LHC and a future 100 TeV proton-proton collider. We make use of invariant mass distributions to enhance the sensitivity on the EFT coefficients and give a first assessment of the impact of jet substructure techniques on the results. The range of validity of the EFT description is estimated, as required to consistently exploit the high-energy range of distributions, pointing out the potential relevance of dimension-8 operators. Our analysis contains a few important improvements over previous studies and identifies some inaccuracies there appearing in connection with the estimate of signal and background rates. The estimated precision on the Higgs trilinear coupling that follows from our results is less optimistic than previously claimed in the literature. We find that a similar to 30% accuracy can be reached on the trilinear coupling at a future 100 TeV collider with 3 ab(-1). Only an O(1) determination instead seems possible at the LHC with the same amount of integrated luminosity.

Amer Physical Soc2015

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Aspects of two dimensional magnetic Schrödinger operatorsquantum hall systems and magnetic Stark resonances

Christian Ferrari

EPFL2003