Twisted Pseudodifferential Calculus and Application to the Quantum Evolution of Molecules

Twisted Pseudodifferential Calculus and Application to the Quantum Evolution of Molecules PDF

Author: AndrŽ Martinez

Publisher: American Mathematical Soc.

Published: 2009-06-05

Total Pages: 96

ISBN-13: 082184296X

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The authors construct an abstract pseudodifferential calculus with operator-valued symbol, suitable for the treatment of Coulomb-type interactions, and they apply it to the study of the quantum evolution of molecules in the Born-Oppenheimer approximation, in the case of the electronic Hamiltonian admitting a local gap in its spectrum. In particular, they show that the molecular evolution can be reduced to the one of a system of smooth semiclassical operators, the symbol of which can be computed explicitely. In addition, they study the propagation of certain wave packets up to long time values of Ehrenfest order.

Effective Hamiltonians for Constrained Quantum Systems

Effective Hamiltonians for Constrained Quantum Systems PDF

Author: Jakob Wachsmuth

Publisher: American Mathematical Soc.

Published: 2014-06-05

Total Pages: 96

ISBN-13: 0821894897

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The authors consider the time-dependent Schrödinger equation on a Riemannian manifold with a potential that localizes a certain subspace of states close to a fixed submanifold . When the authors scale the potential in the directions normal to by a parameter , the solutions concentrate in an -neighborhood of . This situation occurs for example in quantum wave guides and for the motion of nuclei in electronic potential surfaces in quantum molecular dynamics. The authors derive an effective Schrödinger equation on the submanifold and show that its solutions, suitably lifted to , approximate the solutions of the original equation on up to errors of order at time . Furthermore, the authors prove that the eigenvalues of the corresponding effective Hamiltonian below a certain energy coincide up to errors of order with those of the full Hamiltonian under reasonable conditions.

Center Manifolds for Semilinear Equations with Non-Dense Domain and Applications to Hopf Bifurcation in Age Structured Models

Center Manifolds for Semilinear Equations with Non-Dense Domain and Applications to Hopf Bifurcation in Age Structured Models PDF

Author: Pierre Magal

Publisher: American Mathematical Soc.

Published: 2009

Total Pages: 84

ISBN-13: 0821846531

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Several types of differential equations, such as delay differential equations, age-structure models in population dynamics, evolution equations with boundary conditions, can be written as semilinear Cauchy problems with an operator which is not densely defined in its domain. The goal of this paper is to develop a center manifold theory for semilinear Cauchy problems with non-dense domain. Using Liapunov-Perron method and following the techniques of Vanderbauwhede et al. in treating infinite dimensional systems, the authors study the existence and smoothness of center manifolds for semilinear Cauchy problems with non-dense domain. As an application, they use the center manifold theorem to establish a Hopf bifurcation theorem for age structured models.

Microlocal Methods in Mathematical Physics and Global Analysis

Microlocal Methods in Mathematical Physics and Global Analysis PDF

Author: Daniel Grieser

Publisher: Springer Science & Business Media

Published: 2012-12-13

Total Pages: 147

ISBN-13: 3034804660

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Microlocal analysis is a field of mathematics that was invented in the mid-20th century for the detailed investigation of problems from partial differential equations, which incorporated and made rigorous many ideas that originated in physics. Since then it has grown to a powerful machine which is used in global analysis, spectral theory, mathematical physics and other fields, and its further development is a lively area of current mathematical research. In this book extended abstracts of the conference 'Microlocal Methods in Mathematical Physics and Global Analysis', which was held at the University of Tübingen from the 14th to the 18th of June 2011, are collected.​

Lyapunov Exponents and Invariant Manifolds for Random Dynamical Systems in a Banach Space

Lyapunov Exponents and Invariant Manifolds for Random Dynamical Systems in a Banach Space PDF

Author: Zeng Lian

Publisher: American Mathematical Soc.

Published: 2010

Total Pages: 119

ISBN-13: 0821846566

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The authors study the Lyapunov exponents and their associated invariant subspaces for infinite dimensional random dynamical systems in a Banach space, which are generated by, for example, stochastic or random partial differential equations. The authors prove a multiplicative ergodic theorem and then use this theorem to establish the stable and unstable manifold theorem for nonuniformly hyperbolic random invariant sets.

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