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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...

Essential self-adjointness for magnetic Schrödinger operators on non-compact manifolds

Mikhail Shubin (1998/1999)

Séminaire Équations aux dérivées partielles

We give a condition of essential self-adjointness for magnetic Schrödinger operators on non-compact Riemannian manifolds with a given positive smooth measure which is fixed independently of the metric. This condition is related to the classical completeness of a related classical hamiltonian without magnetic field. The main result generalizes the result by I. Oleinik [29,30,31], a shorter and more transparent proof of which was provided by the author in [41]. The main idea, as in [41], consists...

Germes de configurations legendriennes stables et fonctions d'Airy-Weber généralisées

Nguyen Hu'u Du'c, Frédéric Pham (1991)

Annales de l'institut Fourier

On sait depuis Maslov, Arnold, etc... associer à presque tout germe de variété lagrangienne ou legendrienne lisse une classe de fonctions oscillantes qui sous des hypothèses génériques à la Thom fournissent des modèles universels pour le comportement d’une onde lumineuse au voisinage de la caustique.Le présent article étend cette construction à une classe de situations où la variété caractéristique est un germe singulier (union de composantes lisses), qui peut néanmoins être stable en ce sens que...

Hybrid matrix models and their population dynamic consequences

Sanyi Tang (2010)

ESAIM: Mathematical Modelling and Numerical Analysis

In this paper, the main purpose is to reveal what kind of qualitative dynamical changes a continuous age-structured model may undergo as continuous reproduction is replaced with an annual birth pulse. Using the discrete dynamical system determined by the stroboscopic map we obtain an exact periodic solution of system with density-dependent fertility and obtain the threshold conditions for its stability. We also present formal proofs of the supercritical flip bifurcation at the bifurcation as...

Hybrid matrix models and their population dynamic consequences

Sanyi Tang (2003)

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique

In this paper, the main purpose is to reveal what kind of qualitative dynamical changes a continuous age-structured model may undergo as continuous reproduction is replaced with an annual birth pulse. Using the discrete dynamical system determined by the stroboscopic map we obtain an exact periodic solution of system with density-dependent fertility and obtain the threshold conditions for its stability. We also present formal proofs of the supercritical flip bifurcation at the bifurcation as well...

On multivortex solutions in Chern-Simons gauge theory

Michael Struwe, Gabriella Tarantello (1998)

Bollettino dell'Unione Matematica Italiana

Motivati dall'analisi asintotica dei vortici nella teoria di Chern-Simons-Higgs, si studia l'equazione - Δ u = λ e u Ω e u d x - 1 Ω , u H 1 Ω dove Ω = R 2 / Z 2 é il toro piatto bidimensionale. In contrasto con l'analogo problema di Dirichlet, si dimostra che per λ 8 π , 4 π 2 l'equazione ammette una soluzione non banale. Tale soluzione cattura il carattere bidimensionale dell'equazione, nel senso che, per tali valori di λ , l'equazione non può ammettere soluzioni (periodiche) non banali dipendenti da una sola variabile (vedi [10]).

Regularity of the multi-configuration time-dependent Hartree approximation in quantum molecular dynamics

Othmar Koch, Christian Lubich (2007)

ESAIM: Mathematical Modelling and Numerical Analysis

We discuss the multi-configuration time-dependent Hartree (MCTDH) method for the approximation of the time-dependent Schrödinger equation in quantum molecular dynamics. This method approximates the high-dimensional nuclear wave function by a linear combination of products of functions depending only on a single degree of freedom. The equations of motion, obtained via the Dirac-Frenkel time-dependent variational principle, consist of a coupled system of low-dimensional nonlinear partial differential...

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