Semiclassical Theory of Shape Resonances in Quantum Mechanics

Semiclassical Theory of Shape Resonances in Quantum Mechanics PDF

Author: Peter D. Hislop

Publisher: American Mathematical Soc.

Published: 1989

Total Pages: 133

ISBN-13: 0821824627

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In this paper, we prove the existence of shape resonances in the semi-classical approximation for Hamiltonians of the form [italic]H([lowercase Greek]Lambda) [triple bar symbol] −([capital Greek]Delta + ([lowercase Greek]Lambda2[italic]V + [italic]U on [italic]L2([bold]R[superscript italic]n), where ([lowercase Greek]Lambda [triple bar symbol] 1/h[with stroke], h[with stroke] [triple bar symbol] (2[lowercase Greek]Pi)−1h.

Mathematical Theory of Scattering Resonances

Mathematical Theory of Scattering Resonances PDF

Author: Semyon Dyatlov

Publisher: American Mathematical Soc.

Published: 2019-09-10

Total Pages: 634

ISBN-13: 147044366X

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Scattering resonances generalize bound states/eigenvalues for systems in which energy can scatter to infinity. A typical resonance has a rate of oscillation (just as a bound state does) and a rate of decay. Although the notion is intrinsically dynamical, an elegant mathematical formulation comes from considering meromorphic continuations of Green's functions. The poles of these meromorphic continuations capture physical information by identifying the rate of oscillation with the real part of a pole and the rate of decay with its imaginary part. An example from mathematics is given by the zeros of the Riemann zeta function: they are, essentially, the resonances of the Laplacian on the modular surface. The Riemann hypothesis then states that the decay rates for the modular surface are all either or . An example from physics is given by quasi-normal modes of black holes which appear in long-time asymptotics of gravitational waves. This book concentrates mostly on the simplest case of scattering by compactly supported potentials but provides pointers to modern literature where more general cases are studied. It also presents a recent approach to the study of resonances on asymptotically hyperbolic manifolds. The last two chapters are devoted to semiclassical methods in the study of resonances.

Spectral Theory and Mathematical Physics: A Festschrift in Honor of Barry Simon's 60th Birthday

Spectral Theory and Mathematical Physics: A Festschrift in Honor of Barry Simon's 60th Birthday PDF

Author: Fritz Gesztesy

Publisher: American Mathematical Soc.

Published: 2007

Total Pages: 528

ISBN-13: 082184248X

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This Festschrift had its origins in a conference called SimonFest held at Caltech, March 27-31, 2006, to honor Barry Simon's 60th birthday. It is not a proceedings volume in the usual sense since the emphasis of the majority of the contributions is on reviews of the state of the art of certain fields, with particular focus on recent developments and open problems. The bulk of the articles in this Festschrift are of this survey form, and a few review Simon's contributions to aparticular area. Part 1 contains surveys in the areas of Quantum Field Theory, Statistical Mechanics, Nonrelativistic Two-Body and $N$-Body Quantum Systems, Resonances, Quantum Mechanics with Electric and Magnetic Fields, and the Semiclassical Limit. Part 2 contains surveys in the areas of Random andErgodic Schrodinger Operators, Singular Continuous Spectrum, Orthogonal Polynomials, and Inverse Spectral Theory. In several cases, this collection of surveys portrays both the history of a subject and its current state of the art. A substantial part of the contributions to this Festschrift are survey articles on the state of the art of certain areas with special emphasis on open problems. This will benefit graduate students as well as researchers who want to get a quick, yet comprehensiveintroduction into an area covered in this volume.

Direct and Inverse Problems

Direct and Inverse Problems PDF

Author: Boris N. Zakhariev

Publisher: Springer Science & Business Media

Published: 2012-12-06

Total Pages: 234

ISBN-13: 3642956157

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Rapid progress in quantum theory brings us new important results which are often not immediately clear to all who need them. But fortunately, this is also followed by simplifications and unifications of our previous concepts. The inverse problem method ("The most beautiful idea of the XX-th century" - Zakharov et aI., 1980) has just both these aspects. It is rather astonishing that it took 50 years after the foundation of quantum mechanics for the creation of the "pictures" showing the direct connection of obser vables with interactions. Recently, illustrations of this type began to appear in the literature (e. g., how potentials are deformed with thc shift of one energy level or change of some resonance reduced width). Although they are transparent to those studying the quantum world and can be included within the necessary elements of quantum literacy, they are still largely unknown even to many specialists. For the first time, the most interesting of these pictures enriching our quantum intuition are col lected here and placed at your disposal. The readers of this monograph have the advantage of getting the latest information which became available after the publication of the Russian edition. It has been incor porated here in the simplest presentation possible. For example, new sections con cerning exactly solvable models, including the multi-channel, multi-dimensional ones and with time dependent potentials have been added. The first attempts in solving the three-body inverse problem are also mentioned.

Long Time Behaviour of Classical and Quantum Systems

Long Time Behaviour of Classical and Quantum Systems PDF

Author: Sandro Graffi

Publisher: World Scientific

Published: 2001

Total Pages: 320

ISBN-13: 9789812794598

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Return to equilibrium in classical and quantum systems / Carlangelo Liverani -- Quantum resonances and trapped trajectories / Johannes Sjostrand -- Return to thermal equilibrium in quantum statistical mechanics / Volker Bach -- Small oscillations in some nonlinear PDE's / Dario Bambusi and Simone Paleari -- The semi-classical Van-Vleck Formula. Application to the Aharonov-Bohm effect / Jean-Marie Bily and Didier Robert -- Fractal dimensions and quantum evolution associated with sparse potential Jacobi matrices / Jean-Michel Combes and Giorgio Mantica -- Infinite step billiards / Mirko Degli Esposti -- Semiclassical expansion for the thermodynamic limit of the ground state energy of Kac's operator / Bernard Helffer and Thierry Ramond -- Asymptotics of scattering poles for two strictly convex obstacles / Mitsuru Ikawa -- Parabolic dynamical systems and inducing / Stefano Isola -- QFT for scalar particles in external fields on Riemannian manifolds / Hiroshi Isozaki -- Existence and born-Oppenheimer Asymptotics of the total scattering cross-section in ion-atom collisions / Thierry Jecko, Markus Klein and Xue Ping Wang -- On asymptotic perturbation theory for quantum mechanics / Gheorghe Nenciu -- Destruction of the beating effect in a periodically driven double-well / Andrea Sacchetti -- Berezin-Toeplitz quantization and Berezin transform / Martin Schlichenmaier.

Introduction to Spectral Theory

Introduction to Spectral Theory PDF

Author: P.D. Hislop

Publisher: Springer Science & Business Media

Published: 2012-12-06

Total Pages: 331

ISBN-13: 146120741X

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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 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 decade. This book is designed to be used as a textbook, unlike the competitors which are either too fundamental in their approach or are too abstract in nature to be considered as texts. The authors' text fills a gap in the marketplace.

Long Time Behaviour Of Classical And Quantum Systems - Proceedings Of The Bologna Aptex International Conference

Long Time Behaviour Of Classical And Quantum Systems - Proceedings Of The Bologna Aptex International Conference PDF

Author: Sandro Graffi

Publisher: World Scientific

Published: 2001-04-02

Total Pages: 299

ISBN-13: 9814491535

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This book is centered on the two minicourses conducted by C Liverani (Rome) and J Sjoestrand (Paris) on the return to equilibrium in classical statistical mechanics and the location of quantum resonances via semiclassical analysis, respectively. The other contributions cover related topics of classical and quantum mechanics, such as scattering theory, classical and quantum statistical mechanics, dynamical localization, quantum chaos, ergodic theory and KAM techniques.

Coherent State Limit of Cavity Vibrations In Analog To Quantum Acoustics

Coherent State Limit of Cavity Vibrations In Analog To Quantum Acoustics PDF

Author: professor ibrahim elnoshokaty

Publisher: ibrahim elnoshokaty

Published:

Total Pages: 429

ISBN-13:

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The book describes the author's research on the quantum behavior of mechanical oscillators in the form of surface acoustic waves (SAWs). The text provides a historical perspective on the use of mechanical oscillators in technological advancements, from pendulum clocks to the development of quantum technologies. The author highlights the unique advantages of SAW systems in the development of quantum technologies and the lessons that can be learned from this particular platform. The text also discusses the potential applications of SAW systems in the development of quantum technologies. The author emphasizes the importance of understanding the quantum behavior of mechanical systems and the potential for these systems to drive scientific advancement. The text discusses the quantum behavior of mechanical oscillators in the form of surface acoustic waves (SAWs) and their potential applications in quantum technologies. The author highlights the unique advantages of SAW systems and the lessons that can be learned from this platform. The text also discusses the history of mechanical oscillators in technological advancements, from pendulum clocks to the development of quantum technologies.The given input text describes the author's research on the quantum behavior of mechanical oscillators in the form of surface acoustic waves (SAWs). The text provides a historical perspective on the use of mechanical oscillators in technological advancements, from pendulum clocks to the development of quantum technologies. The author highlights the unique advantages of SAW systems in the development of quantum technologies and the lessons that can be learned from this particular platform. The text also discusses the potential applications of SAW systems in the development of quantum technologies. The author emphasizes the importance of understanding the quantum behavior of mechanical systems and the potential for these systems to drive scientific advancement. The input text describes the author's research on the quantum behavior of mechanical oscillators in the form of surface acoustic waves (SAWs). The text provides a historical perspective on the use of mechanical oscillators in technological advancements, from pendulum clocks to the development of quantum technologies. The author highlights the unique advantages of SAW systems in the development of quantum technologies and the lessons that can be learned from this platform. The text also discusses the potential applications of SAW systems in the development of quantum technologies. The author emphasizes the importance of understanding the quantum behavior of mechanical systems and the potential for these systems to drive scientific advancement. Research Highlights: * Quantum acoustics behavior of mechanical oscillators in SAWs discussed * Historical perspective on mechanical oscillators in technological advancements * Potential applications of SAW systems in quantum acoustics a technologies discussed * Importance of understanding quantum behavior of mechanical systems emphasized Key word Acoustic wave, Surface acoustic wave, SAW, Mechanical oscillator, Quantum mechanics, Quantum technology

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