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Saturday, July 18, 2020 | History

7 edition of Semi-classical analysis for the Schrödinger operator and applications found in the catalog.

Semi-classical analysis for the Schrödinger operator and applications

by Bernard Helffer

  • 244 Want to read
  • 16 Currently reading

Published by Springer-Verlag in Berlin, New York .
Written in English

    Subjects:
  • Schrödinger operator.,
  • Differential equations, Partial -- Asymptotic theory.,
  • Spectral theory (Mathematics)

  • Edition Notes

    StatementBernard Helffer.
    SeriesLecture notes in mathematics ;, 1336, Lecture notes in mathematics (Springer-Verlag) ;, 1336.
    Classifications
    LC ClassificationsQA3 .L28 no. 1336a, QA329 .L28 no. 1336a
    The Physical Object
    Paginationiv, 107 p. :
    Number of Pages107
    ID Numbers
    Open LibraryOL2136164M
    ISBN 103540500766, 0387500766
    LC Control Number88202680


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Semi-classical analysis for the Schrödinger operator and applications by Bernard Helffer Download PDF EPUB FB2

This introduction to semi-classical analysis is an extension of a course given by the author at the University of Nankai. It presents for some of the standard cases presented in quantum mechanics books a rigorous study of the tunneling effect, as an Cited by: This introduction to semi-classical analysis is an extension of a course given by the author at the University of Nankai.

It presents for some of the standard cases presented in quantum mechanics books a rigorous study of the tunneling effect, as an introduction to recent research work. The book.

About this book. Introduction. This introduction to semi-classical analysis is an extension of a course given by the author at the University of Nankai. It presents for some of the standard cases presented in quantum mechanics books a rigorous study of the tunneling effect, as an introduction to recent research work.

Generalities on semi-classical analysis.- B.K.W. Construction for a potential near the bottom in the case of non-degenerate minima.- The decay of the eigenfunctions.- Study of interaction between the wells.- An introduction to recent results of Witten.- On Schroedinger operators with periodic electric potentials The book consists of two relatively independent parts: WKB analysis, and caustic crossing.

In the first part, the basic linear WKB theory is constructed and then extended to the nonlinear framework. The most difficult supercritical case is discussed in detail, together with some of its consequences concerning instability by: The main point is the analysis of the asymptotic behavior, in the semi-classical sense, of the ground state energy for the Schrödinger operator with a magnetic field.

We consider the case when the locus of the minima of the intensity of the magnetic field is compact and our study is sharper when this locus is an hypersurface Cited by: 6. The intention of this book is to introduce students to active areas of research in mathematical physics in a rather direct way minimizing the use of abstract mathematics.

The main features are geometric methods in spectral analysis, exponential decay of eigenfunctions, semi-classical analysis of bound state problems, and semi-classical analysis of : Paperback. Abstract.

We study the asymptotic behavior, in a “semi-classical limit,” of the first eigenvalues (i.e., the groundstate energies) of a class of Schrödinger operators with magnetic fields and the relationship of this behavior with compactness in the -Neumann problem on Hartogs domains by: Applications of WKB analysis to functional analysis, in particular to the Cauchy problem for nonlinear Schrödinger equations, are also given.

In the second part, caustic crossing is described, especially when the caustic is reduced to a point, and the link with nonlinear scattering operators. Abstract. In this lecture 1 we present some survey on the semiclassical analysis of the Schrödinger operator with magnetic fields with emphasis on the recent results by mery [] and extensions obtained in collaboration with d [].The main point is the analysis of the asymptotic behavior, in the semi-classical sense, of the ground state energy for the Schrödinger operator.

We study the asymptotic behavior, in a “semi-classical limit,” of the first eigenvalues (i.e., the groundstate energies) of a class of Schrödinger operators with magnetic fields and the relationship of this behavior with compactness in the ∂-Neumann problem on.

Semi-classical analysis for magnetic Schr¨odinger operators and applications Bernard Helffer (Univ Paris-Sud and CNRS) McGill, May Bernard Helffer (Univ Paris-Sud and CNRS) Semi-classical analysis for magnetic Schrodinger operators and applications.

Introduction Main goals Rough results Two-terms asymptotics in the case of a. semi-classical. There is a huge litterature on the counting function and connected spectral quantities. We only look in this talk at the bottom. Bernard Hel er (Univ Paris-Sud and CNRS) Semi-classical analysis for magnetic Schr odinger operators and applications: old and new.

Conference in honor of M. Shubin for his 65 birthday. Applications of WKB analysis to functional analysis, in particular to the Cauchy problem for nonlinear Schrödinger equations, are also given.

In the second part, caustic crossing is described, especially when the caustic is reduced to a point, and the link with nonlinear scattering operators. This book is a brief, but self-contained, introduction to the mathematical methods of quantum mechanics, with a view towards applications to Schrodinger operators.

Part 1 of the book is a concise introduction to the spectral theory of unbounded operators. Only those topics that will be needed for later applications are : Gerald Teschl. Semi-classical analysis for nonlinear Schrödinger equations in the light of both scattering theory and semi-classical analysis.

two-body Schrodinger operator P =. Helffer B. () Generalities on semi-classical analysis. In: Semi-Classical Analysis for the Schrödinger Operator and Applications.

Lecture Notes in Mathematics, vol   Helffer B. () On Schrödinger operators with magnetic fields. In: Semi-Classical Analysis for the Schrödinger Operator and Applications.

Lecture Notes in Mathematics, vol This monograph serves as a much-needed, self-contained reference on the topic of modulation spaces.

By gathering together state-of-the-art developments and previously unexplored applications, readers will be motivated to make effective use of this topic in future research. The main features are geometric methods in spectral analysis, exponential decay of eigenfunctions, semi-classical analysis of bound state problems, and semi-classical analysis of resonance.

A new geometric point of view along with new techniques are brought out in this book which have both been discovered within the past : Paperback. Helffer B. () On Schrödinger operators with periodic electric potentials. In: Semi-Classical Analysis for the Schrödinger Operator and Applications.

Lecture Notes in Mathematics, vol   Bernard Helffer: Spectral theory and semi-classical analysis for the complex Schrödinger operator We consider the operator Ah=−h2Δ+iV in the semi-classical limit h→0, where V is a smooth.the (quantum) Hamiltonian,ortheSchr¨odinger operator. Itisalwaysas-sumed that H does not depend explicitly on time.

Axiom There exists a one parameter group U t of unitary operators (evolution operator) that map an initial state ψ 0 at the time t =0to the state ψ(t)=U tψ 0 at the time t. The operator U t is of the form () U t = e.