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Edge-reinforced random walk (ERRW), introduced by Coppersmith and Diaconis in 1986 [8], is a random process which takes values in the vertex set of a graph and is more likely to cross edges it has visited before. We show that it can be represented in terms of a vertex-reinforced jump process (VRJP) with independent gamma conductances; the VRJP was conceived by Werner and first studied by Davis and Volkov [10, 11], and is a continuous-time process favouring sites with more local time. We calculate,...
We recount here some preliminary attempts to devise quantum analogues of certain aspects of Mather’s theory of minimizing measures [M1-2, M-F], augmented by the PDE theory from Fathi [F1,2] and from [E-G1]. This earlier work provides us with a Lipschitz continuous function solving the eikonal equation aėȧnd a probability measure solving a related transport equation.We present some elementary formal identities relating certain quantum states and . We show also how to build out of an approximate...
We study stationary solutions of the damped wave equation on a compact and smooth Riemannian manifold without boundary. In the high frequency limit, we prove that a sequence of -damped stationary solutions cannot be completely concentrated in small neighborhoods of a small fixed hyperbolic subset made of -damped trajectories of the geodesic flow.The article also includes an appendix (by S. Nonnenmacher and the author) where we establish the existence of an inverse logarithmic strip without eigenvalues...
We consider the Pauli operator selfadjoint in , . Here , , are the Pauli matrices, is the magnetic potential, is the coupling constant, and is the electric potential which decays at infinity. We suppose that the magnetic field generated by satisfies some regularity conditions; in particular, its norm is lower-bounded by a positive constant, and, in the case , its direction is constant. We investigate the asymptotic behaviour as of the number of the eigenvalues of smaller than...
The electronic structure of the nanocylinder is investigated. Two cases of this kind of the nanostructure are explored: the defect-free nanocylinder and the nanocylinder whose geometry is perturbed by 2 heptagonal defects lying on the opposite sides. The characteristic quantity which is of our interest is the local density of states. To calculate it, the continuum gauge field-theory model will be used. In this model, the Dirac-like equation is solved on a curved surface. This procedure was used...
We describe a simple linear algebra idea which has been used in different branches of mathematics such as bifurcation theory, partial differential equations and numerical analysis. Under the name of the Schur complement method it is one of the standard tools of applied linear algebra. In PDE and spectral analysis it is sometimes called the Grushin problem method, and here we concentrate on its uses in the study of infinite dimensional problems, coming from partial differential operators of mathematical...
In this article, we give a necessary and sufficient condition in the perturbation regime on the existence of eigenvalues embedded between two thresholds. For an eigenvalue of the unperturbed operator embedded at a threshold, we prove that it can produce both discrete eigenvalues and resonances. The locations of the eigenvalues and resonances are given.
L’objet de cet exposé est de montrer comment l’évolution de Schrödinger pour le problème à corps quantique est approchée, lorsque tend vers l’infini, dans un régime convenable, par une évolution non-linéaire en dimension trois d’espace. On traitera le cas des bosons, qui conduit à l’équation de Schrödinger-Poisson, et celui des fermions, qui débouche sur le système de Hartree-Fock.
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