Bernard Helffer's Books

Hypoelliptic Estimates and Spectral Theory for Fokker-Planck Operators and Witten Laplacians

By: Francis Nier,Bernard Helffer

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Hypoelliptic Estimates and Spectral Theory for Fokker-Planck Operators and Witten Laplacians

By: Francis Nier,Bernard Helffer

There has recently been a renewal of interest in Fokker-Planck operators, motivated by problems in statistical physics, in kinetic equations, and differential geometry. Compared to more standard problems in the spectral theory of partial differential operators, those operators are not self-adjoint and only hypoelliptic. The aim of the analysis is to give, as generally as possible, an accurate qualitative and quantitative description of the exponential return to the thermodynamical equilibrium. While exploring and improving recent results in this direction, this volume proposes a review of known techniques on: the hypoellipticity of polynomial of vector fields and its global counterpart, the global Weyl-Hörmander pseudo-differential calculus, the spectral theory of non-self-adjoint operators, the semi-classical analysis of Schrödinger-type operators, the Witten complexes, and the Morse inequalities.

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Page Count

228

Publisher

Springer Science & Business Media

Published Date

ISBN 10

3540242007

ISBN 13

9783540242000

Semi-Classical Analysis for the Schrödinger Operator and Applications

By: Bernard Helffer

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Semi-Classical Analysis for the Schrödinger Operator and Applications

By: Bernard Helffer

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 may be read by a graduate student familiar with the classic book of Reed-Simon, and for some chapters basic notions in differential geometry. The mathematician will find here a nice application of PDE techniques and the physicist will discover the precise link between approximate solutions (B.K.W. constructions) and exact eigenfunctions (in every dimension). An application to Witten's approach for the proof of the Morse inequalities is given, as are recent results for the Schrödinger operator with periodic potentials.

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Page Count

113

Publisher

Springer

Published Date

ISBN 10

3540459138

ISBN 13

9783540459132

Semiclassical Analysis, Witten Laplacians, And Statistical Mechanics

By: Bernard Helffer

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Semiclassical Analysis, Witten Laplacians, And Statistical Mechanics

By: Bernard Helffer

This important book explains how the technique of Witten Laplacians may be useful in statistical mechanics. It considers the problem of analyzing the decay of correlations, after presenting its origin in statistical mechanics. In addition, it compares the Witten Laplacian approach with other techniques, such as the transfer matrix approach and its semiclassical analysis. The author concludes by providing a complete proof of the uniform Log-Sobolev inequality.

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Page Count

190

Publisher

World Scientific

Published Date

ISBN 10

9814487902

ISBN 13

9789814487900

Spectral Theory and Its Applications

By: Bernard Helffer

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Spectral Theory and Its Applications

By: Bernard Helffer

Introduces the basic tools in spectral analysis using numerous examples from the Schrödinger operator theory and various branches of physics.

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Page Count

263

Publisher

Cambridge University Press

Published Date

ISBN 10

110703230X

ISBN 13

9781107032309

Pseudo-Differential Operators

Quantization and Signals

By: Hans G. Feichtinger,Bernard Helffer,Michael Lamoureux,Nicolas Lerner,Joachim Toft

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Pseudo-Differential Operators

Quantization and Signals

By: Hans G. Feichtinger,Bernard Helffer,Michael Lamoureux,Nicolas Lerner,Joachim Toft

Pseudo-differential operators were initiated by Kohn, Nirenberg and Hörmander in the sixties of the last century. Beside applications in the general theory of partial differential equations, they have their roots also in the study of quantization first envisaged by Hermann Weyl thirty years earlier. Thanks to the understanding of the connections of wavelets with other branches of mathematical analysis, quantum physics and engineering, such operators have been used under different names as mathematical models in signal analysis since the last decade of the last century. The volume investigates the mathematics of quantization and signals in the context of pseudo-differential operators, Weyl transforms, Daubechies operators, Wick quantization and time-frequency localization operators. Applications to quantization, signal analysis and the modern theory of PDE are highlighted.

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Page Count

235

Publisher

Springer

Published Date

ISBN 10

3540682686

ISBN 13

9783540682684

Penalising Brownian Paths

By: Bernard Roynette,Marc Yor

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Penalising Brownian Paths

By: Bernard Roynette,Marc Yor

Penalising a process is to modify its distribution with a limiting procedure, thus defining a new process whose properties differ somewhat from those of the original one. We are presenting a number of examples of such penalisations in the Brownian and Bessel processes framework. The Martingale theory plays a crucial role. A general principle for penalisation emerges from these examples. In particular, it is shown in the Brownian framework that a positive sigma-finite measure takes a large class of penalisations into account.

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Page Count

291

Publisher

Springer

Published Date

ISBN 10

3540896996

ISBN 13

9783540896999

Spectral Theory and its Applications

By: Bernard Helffer

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Spectral Theory and its Applications

By: Bernard Helffer

Bernard Helffer's graduate-level introduction to the basic tools in spectral analysis is illustrated by numerous examples from the Schrödinger operator theory and various branches of physics: statistical mechanics, superconductivity, fluid mechanics and kinetic theory. The later chapters also introduce non self-adjoint operator theory with an emphasis on the role of the pseudospectra. The author's focus on applications, along with exercises and examples, enables readers to connect theory with practice so that they develop a good understanding of how the abstract spectral theory can be applied. The final chapter provides various problems that have been the subject of active research in recent years and will challenge the reader's understanding of the material covered.

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Page Count

263

Publisher

Cambridge University Press

Published Date

ISBN 10

1139620517

ISBN 13

9781139620512

Hypoelliptic Estimates and Spectral Theory for Fokker-Planck Operators and Witten Laplacians

By: Francis Nier,Bernard Helffer

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Hypoelliptic Estimates and Spectral Theory for Fokker-Planck Operators and Witten Laplacians

By: Francis Nier,Bernard Helffer

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Page Count

228

Publisher

Springer Science & Business Media

Published Date

ISBN 10

3540242007

ISBN 13

9783540242000

Pseudo-Differential Operators

Quantization and Signals

By: Hans G. Feichtinger,Bernard Helffer,Michael Lamoureux,Nicolas Lerner,Joachim Toft

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Pseudo-Differential Operators

Quantization and Signals

By: Hans G. Feichtinger,Bernard Helffer,Michael Lamoureux,Nicolas Lerner,Joachim Toft

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Solve jigsaw puzzle

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Page Count

235

Publisher

Springer Science & Business Media

Published Date

ISBN 10

354068266X

ISBN 13

9783540682660

Book's by Bernard Helffer

Hypoelliptic Estimates and Spectral Theory for Fokker-Planck Operators and Witten Laplacians

Hypoelliptic Estimates and Spectral Theory for Fokker-Planck Operators and Witten Laplacians

Semi-Classical Analysis for the Schrödinger Operator and Applications

Semi-Classical Analysis for the Schrödinger Operator and Applications

Semiclassical Analysis, Witten Laplacians, And Statistical Mechanics

Semiclassical Analysis, Witten Laplacians, And Statistical Mechanics

Spectral Theory and Its Applications

Spectral Theory and Its Applications

Pseudo-Differential Operators

Pseudo-Differential Operators

Penalising Brownian Paths

Penalising Brownian Paths

Spectral Theory and its Applications

Spectral Theory and its Applications

Hypoelliptic Estimates and Spectral Theory for Fokker-Planck Operators and Witten Laplacians

Hypoelliptic Estimates and Spectral Theory for Fokker-Planck Operators and Witten Laplacians

Pseudo-Differential Operators

Pseudo-Differential Operators

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