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absence de résonance près du réel pour l’opérateur de Schrödinger

Nicolas Burq (1997/1998)

Séminaire Équations aux dérivées partielles

On donne dans cet exposé des bornes inférieures universelles, en limite semiclassique, de la hauteur des résonances de forme associées aux opérateurs de Schrödinger à l’extérieur d’obstacles avec des conditions au bord de Dirichlet ou de Neumann et des potentiels analytiquement dilatables et tendant vers 0 à l’infini. Ces bornes inférieures sont exponentiellement petites par rapport à la constante de Planck.

An error analysis of the multi-configuration time-dependent Hartree method of quantum dynamics

Dajana Conte, Christian Lubich (2010)

ESAIM: Mathematical Modelling and Numerical Analysis

This paper gives an error analysis of the multi-configuration time-dependent Hartree (MCTDH) method for the approximation of multi-particle time-dependent Schrödinger equations. The MCTDH method approximates the multivariate wave function by a linear combination of products of univariate functions and replaces the high-dimensional linear Schrödinger equation by a coupled system of ordinary differential equations and low-dimensional nonlinear partial differential equations. The main result of this...

Analytic solutions to nonlocal abstract equations

Marina Ghisi (2003)

Bollettino dell'Unione Matematica Italiana

In this paper we study the problem of existence of global solutions for some classes of abstract equations, that generalize some type of Klein-Gordon equations, with nonlinear nonlocal terms of Kirchhoff type. We find some conditions that guarantee the existence of such solutions whether in presence or in absence of a conserved energy.

Asymptotic behavior of solutions to Schrödinger equations near an isolated singularity of the electromagnetic potential

Veronica Felli, Alberto Ferrero, Susanna Terracini (2011)

Journal of the European Mathematical Society

Asymptotics of solutions to Schrödinger equations with singular magnetic and electric potentials is investigated. By using a Almgren type monotonicity formula, separation of variables, and an iterative Brezis–Kato type procedure, we describe the exact behavior near the singularity of solutions to linear and semilinear (critical and subcritical) elliptic equations with an inverse square electric potential and a singular magnetic potential with a homogeneity of order −1.

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