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"...A graduate level text introducing readers to semiclassical and microlocal methods in PDE." -- from xi.
Book Synopsis Semiclassical Analysis by : Maciej Zworski
Download or read book Semiclassical Analysis written by Maciej Zworski and published by American Mathematical Soc.. This book was released on 2012 with total page 448 pages. Available in PDF, EPUB and Kindle. Book excerpt: "...A graduate level text introducing readers to semiclassical and microlocal methods in PDE." -- from xi.
This book presents the techniques used in the microlocal treatment of semiclassical problems coming from quantum physics in a pedagogical, way and is mainly addressed to non-specialists in the subject. It is based on lectures taught by the author over several years, and includes many exercises providing outlines of useful applications of the semi-classical theory.
Book Synopsis An Introduction to Semiclassical and Microlocal Analysis by : André Bach
Download or read book An Introduction to Semiclassical and Microlocal Analysis written by André Bach and published by Springer Science & Business Media. This book was released on 2013-03-14 with total page 193 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book presents the techniques used in the microlocal treatment of semiclassical problems coming from quantum physics in a pedagogical, way and is mainly addressed to non-specialists in the subject. It is based on lectures taught by the author over several years, and includes many exercises providing outlines of useful applications of the semi-classical theory.
Book Synopsis Semi-classical Analysis by : Victor Guillemin
Download or read book Semi-classical Analysis written by Victor Guillemin and published by . This book was released on 2013 with total page 446 pages. Available in PDF, EPUB and Kindle. Book excerpt:
These lecture notes review recent results on the high-frequency analysis of nonlinear Schrödinger equations in the presence of an external potential. 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 phenomena. 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 is investigated.These notes are self-contained and combine selected articles written by the author over the past ten years in a coherent manner, with some simplified proofs. Examples and figures are provided to support the intuition, and comparisons with other equations such as the nonlinear wave equation are provided.
Book Synopsis Semi-classical Analysis For Nonlinear Schrodinger Equations by : Remi Carles
Download or read book Semi-classical Analysis For Nonlinear Schrodinger Equations written by Remi Carles and published by World Scientific. This book was released on 2008-03-04 with total page 256 pages. Available in PDF, EPUB and Kindle. Book excerpt: These lecture notes review recent results on the high-frequency analysis of nonlinear Schrödinger equations in the presence of an external potential. 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 phenomena. 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 is investigated.These notes are self-contained and combine selected articles written by the author over the past ten years in a coherent manner, with some simplified proofs. Examples and figures are provided to support the intuition, and comparisons with other equations such as the nonlinear wave equation are provided.
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.
Book Synopsis Semiclassical Analysis, Witten Laplacians, and Statistical Mechanics by : Bernard Helffer
Download or read book Semiclassical Analysis, Witten Laplacians, and Statistical Mechanics written by Bernard Helffer and published by World Scientific. This book was released on 2002 with total page 200 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.
Book Synopsis Semiclassical Analysis for Diffusions and Stochastic Processes by : Vasiliĭ Nikitich Kolokolʹt︠s︡ov
Download or read book Semiclassical Analysis for Diffusions and Stochastic Processes written by Vasiliĭ Nikitich Kolokolʹt︠s︡ov and published by . This book was released on 2000 with total page 0 pages. Available in PDF, EPUB and Kindle. Book excerpt:
This book is an excellent, comprehensive introduction to semiclassical analysis. I believe it will become a standard reference for the subject. —Alejandro Uribe, University of Michigan Semiclassical analysis provides PDE techniques based on the classical-quantum (particle-wave) correspondence. These techniques include such well-known tools as geometric optics and the Wentzel–Kramers–Brillouin approximation. Examples of problems studied in this subject are high energy eigenvalue asymptotics and effective dynamics for solutions of evolution equations. From the mathematical point of view, semiclassical analysis is a branch of microlocal analysis which, broadly speaking, applies harmonic analysis and symplectic geometry to the study of linear and nonlinear PDE. The book is intended to be a graduate level text introducing readers to semiclassical and microlocal methods in PDE. It is augmented in later chapters with many specialized advanced topics which provide a link to current research literature.
Book Synopsis Semiclassical Analysis by : Maciej Zworski
Download or read book Semiclassical Analysis written by Maciej Zworski and published by American Mathematical Society. This book was released on 2022-05-09 with total page 431 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book is an excellent, comprehensive introduction to semiclassical analysis. I believe it will become a standard reference for the subject. —Alejandro Uribe, University of Michigan Semiclassical analysis provides PDE techniques based on the classical-quantum (particle-wave) correspondence. These techniques include such well-known tools as geometric optics and the Wentzel–Kramers–Brillouin approximation. Examples of problems studied in this subject are high energy eigenvalue asymptotics and effective dynamics for solutions of evolution equations. From the mathematical point of view, semiclassical analysis is a branch of microlocal analysis which, broadly speaking, applies harmonic analysis and symplectic geometry to the study of linear and nonlinear PDE. The book is intended to be a graduate level text introducing readers to semiclassical and microlocal methods in PDE. It is augmented in later chapters with many specialized advanced topics which provide a link to current research literature.
The book gives a streamlined introduction to quantum mechanics while describing the basic mathematical structures underpinning this discipline. Starting with an overview of key physical experiments illustrating the origin of the physical foundations, the book proceeds with a description of the basic notions of quantum mechanics and their mathematical content. It then makes its way to topics of current interest, specifically those in which mathematics plays an important role. The more advanced topics presented include: many-body systems, modern perturbation theory, path integrals, the theory of resonances, adiabatic theory, geometrical phases, Aharonov-Bohm effect, density functional theory, open systems, the theory of radiation (non-relativistic quantum electrodynamics), and the renormalization group. With different selections of chapters, the book can serve as a text for an introductory, intermediate, or advanced course in quantum mechanics. Some of the sections could be used for introductions to geometrical methods in Quantum Mechanics, to quantum information theory and to quantum electrodynamics and quantum field theory.
Book Synopsis MATHEMATICAL CONCEPTS OF QUANTUM MECHANICS by : STEPHEN J. GUSTAFSON
Download or read book MATHEMATICAL CONCEPTS OF QUANTUM MECHANICS written by STEPHEN J. GUSTAFSON and published by . This book was released on 2020 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt: The book gives a streamlined introduction to quantum mechanics while describing the basic mathematical structures underpinning this discipline. Starting with an overview of key physical experiments illustrating the origin of the physical foundations, the book proceeds with a description of the basic notions of quantum mechanics and their mathematical content. It then makes its way to topics of current interest, specifically those in which mathematics plays an important role. The more advanced topics presented include: many-body systems, modern perturbation theory, path integrals, the theory of resonances, adiabatic theory, geometrical phases, Aharonov-Bohm effect, density functional theory, open systems, the theory of radiation (non-relativistic quantum electrodynamics), and the renormalization group. With different selections of chapters, the book can serve as a text for an introductory, intermediate, or advanced course in quantum mechanics. Some of the sections could be used for introductions to geometrical methods in Quantum Mechanics, to quantum information theory and to quantum electrodynamics and quantum field theory.
This book presents the basic methods and applications in semiclassical approximation in the light of developments.
Book Synopsis Spectral Asymptotics in the Semi-Classical Limit by : Mouez Dimassi
Download or read book Spectral Asymptotics in the Semi-Classical Limit written by Mouez Dimassi and published by Cambridge University Press. This book was released on 1999-09-16 with total page 243 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book presents the basic methods and applications in semiclassical approximation in the light of developments.
This book represents the first asymptotic analysis, via completely integrable techniques, of the initial value problem for the focusing nonlinear Schrödinger equation in the semiclassical asymptotic regime. This problem is a key model in nonlinear optical physics and has increasingly important applications in the telecommunications industry. The authors exploit complete integrability to establish pointwise asymptotics for this problem's solution in the semiclassical regime and explicit integration for the underlying nonlinear, elliptic, partial differential equations suspected of governing the semiclassical behavior. In doing so they also aim to explain the observed gradient catastrophe for the underlying nonlinear elliptic partial differential equations, and to set forth a detailed, pointwise asymptotic description of the violent oscillations that emerge following the gradient catastrophe. To achieve this, the authors have extended the reach of two powerful analytical techniques that have arisen through the asymptotic analysis of integrable systems: the Lax-Levermore-Venakides variational approach to singular limits in integrable systems, and Deift and Zhou's nonlinear Steepest-Descent/Stationary Phase method for the analysis of Riemann-Hilbert problems. In particular, they introduce a systematic procedure for handling certain Riemann-Hilbert problems with poles accumulating on curves in the plane. This book, which includes an appendix on the use of the Fredholm theory for Riemann-Hilbert problems in the Hölder class, is intended for researchers and graduate students of applied mathematics and analysis, especially those with an interest in integrable systems, nonlinear waves, or complex analysis.
Book Synopsis Semiclassical Soliton Ensembles for the Focusing Nonlinear Schrödinger Equation (AM-154) by : Spyridon Kamvissis
Download or read book Semiclassical Soliton Ensembles for the Focusing Nonlinear Schrödinger Equation (AM-154) written by Spyridon Kamvissis and published by Princeton University Press. This book was released on 2003-08-18 with total page 280 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book represents the first asymptotic analysis, via completely integrable techniques, of the initial value problem for the focusing nonlinear Schrödinger equation in the semiclassical asymptotic regime. This problem is a key model in nonlinear optical physics and has increasingly important applications in the telecommunications industry. The authors exploit complete integrability to establish pointwise asymptotics for this problem's solution in the semiclassical regime and explicit integration for the underlying nonlinear, elliptic, partial differential equations suspected of governing the semiclassical behavior. In doing so they also aim to explain the observed gradient catastrophe for the underlying nonlinear elliptic partial differential equations, and to set forth a detailed, pointwise asymptotic description of the violent oscillations that emerge following the gradient catastrophe. To achieve this, the authors have extended the reach of two powerful analytical techniques that have arisen through the asymptotic analysis of integrable systems: the Lax-Levermore-Venakides variational approach to singular limits in integrable systems, and Deift and Zhou's nonlinear Steepest-Descent/Stationary Phase method for the analysis of Riemann-Hilbert problems. In particular, they introduce a systematic procedure for handling certain Riemann-Hilbert problems with poles accumulating on curves in the plane. This book, which includes an appendix on the use of the Fredholm theory for Riemann-Hilbert problems in the Hölder class, is intended for researchers and graduate students of applied mathematics and analysis, especially those with an interest in integrable systems, nonlinear waves, or complex analysis.