Schrödinger Operators, with Applications to Quantum Mechanics and Global Geometry
Author: Hans Ludwig Cycon
Publisher: Springer
Published: 1987
Total Pages: 344
ISBN-13:
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Multidimensional Periodic Schrödinger Operator
Author: Oktay Veliev
Publisher: Springer Nature
Published:
Total Pages: 420
ISBN-13: 3031490355
DOWNLOAD EBOOK →Multidimensional Periodic Schrödinger Operator
Author: Oktay Veliev
Publisher:
Published: 2019
Total Pages: 326
ISBN-13: 9783030245795
DOWNLOAD EBOOK →This book describes the direct and inverse problems of the multidimensional Schrödinger operator with a periodic potential, a topic that is especially important in perturbation theory, constructive determination of spectral invariants and finding the periodic potential from the given Bloch eigenvalues. It provides a detailed derivation of the asymptotic formulas for Bloch eigenvalues and Bloch functions in arbitrary dimensions while constructing and estimating the measure of the iso-energetic surfaces in the high-energy regime. Moreover, it presents a unique method proving the validity of the Bethe-Sommerfeld conjecture for arbitrary dimensions and arbitrary lattices. Using the perturbation theory constructed, it determines the spectral invariants of the multidimensional operator from the given Bloch eigenvalues. Some of these invariants are explicitly expressed by the Fourier coefficients of the potential, making it possible to determine the potential constructively using Bloch eigenvalues as input data. Lastly, the book presents an algorithm for the unique determination of the potential. This updated second edition includes an additional chapter that specifically focuses on lower-dimensional cases, providing the basis for the higher-dimensional considerations of the chapters that follow.
Topics in the Theory of Schrödinger Operators
Author: Huzihiro Araki
Publisher: World Scientific
Published: 2004
Total Pages: 296
ISBN-13: 9789812562470
DOWNLOAD EBOOK →This invaluable book presents reviews of some recent topics in thetheory of SchrAdinger operators. It includes a short introduction tothe subject, a survey of the theory of the SchrAdinger equation whenthe potential depends on the time periodically, an introduction to theso-called FBI transformation (also known as coherent state expansion)with application to the semi-classical limit of the S-matrix, anoverview of inverse spectral and scattering problems, and a study ofthe ground state of the PauliOCoFierz model with the use of thefunctional integral. The material is accessible to graduate studentsand non-expert researchers."
Schrödinger Operators
Author: Hans L. Cycon
Publisher: Springer Science & Business Media
Published: 1987
Total Pages: 337
ISBN-13: 3540167587
DOWNLOAD EBOOK →Are you looking for a concise summary of the theory of Schrödinger operators? Here it is. Emphasizing the progress made in the last decade by Lieb, Enss, Witten and others, the three authors don’t just cover general properties, but also detail multiparticle quantum mechanics – including bound states of Coulomb systems and scattering theory. This corrected and extended reprint contains updated references as well as notes on the development in the field over the past twenty years.
Spectral Theory of Schrodinger Operators
Author: Rafael del Río
Publisher: American Mathematical Soc.
Published: 2004
Total Pages: 264
ISBN-13: 0821832972
DOWNLOAD EBOOK →This volume gathers the articles based on a series of lectures from a workshop held at the Institute of Applied Mathematics of the National University of Mexico. The aim of the book is to present to a non-specialized audience the basic tools needed to understand and appreciate new trends of research on Schrodinger operator theory. Topics discussed include various aspects of the spectral theory of differential operators, the theory of self-adjoint operators, finite rank perturbations, spectral properties of random Schrodinger operators, and scattering theory for Schrodinger operators. The material is suitable for graduate students and research mathematicians interested in differential operators, in particular, spectral theory of Schrodinger operators.
Geometric and Arithmetic Methods in the Spectral Theory of Multidimensional Periodic Operators
Author: M. M. Skriganov
Publisher: American Mathematical Soc.
Published: 1987
Total Pages: 132
ISBN-13: 9780821831045
DOWNLOAD EBOOK →Topics In The Theory Of Schrodinger Operators
Author: Huzihiro Araki
Publisher: World Scientific
Published: 2004-05-07
Total Pages: 288
ISBN-13: 9814482986
DOWNLOAD EBOOK →This invaluable book presents reviews of some recent topics in the theory of Schrödinger operators. It includes a short introduction to the subject, a survey of the theory of the Schrödinger equation when the potential depends on the time periodically, an introduction to the so-called FBI transformation (also known as coherent state expansion) with application to the semi-classical limit of the S-matrix, an overview of inverse spectral and scattering problems, and a study of the ground state of the Pauli-Fierz model with the use of the functional integral. The material is accessible to graduate students and non-expert researchers.
Perturbation Theory for the Schrödinger Operator with a Periodic Potential
Author: Yulia E. Karpeshina
Publisher: Springer
Published: 2006-11-14
Total Pages: 358
ISBN-13: 3540691561
DOWNLOAD EBOOK →The book is devoted to perturbation theory for the Schrödinger operator with a periodic potential, describing motion of a particle in bulk matter. The Bloch eigenvalues of the operator are densely situated in a high energy region, so regular perturbation theory is ineffective. The mathematical difficulties have a physical nature - a complicated picture of diffraction inside the crystal. The author develops a new mathematical approach to this problem. It provides mathematical physicists with important results for this operator and a new technique that can be effective for other problems. The semiperiodic Schrödinger operator, describing a crystal with a surface, is studied. Solid-body theory specialists can find asymptotic formulae, which are necessary for calculating many physical values.
Many-Body Schrödinger Equation
Author: Hiroshi Isozaki
Publisher: Springer Nature
Published: 2023-08-28
Total Pages: 411
ISBN-13: 9819937043
DOWNLOAD EBOOK →Spectral properties for Schrödinger operators are a major concern in quantum mechanics both in physics and in mathematics. For the few-particle systems, we now have sufficient knowledge for two-body systems, although much less is known about N-body systems. The asymptotic completeness of time-dependent wave operators was proved in the 1980s and was a landmark in the study of the N-body problem. However, many problems are left open for the stationary N-particle equation. Due to the recent rapid development of computer power, it is now possible to compute the three-body scattering problem numerically, in which the stationary formulation of scattering is used. This means that the stationary theory for N-body Schrödinger operators remains an important problem of quantum mechanics. It is stressed here that for the three-body problem, we have a satisfactory stationary theory. This book is devoted to the mathematical aspects of the N-body problem from both the time-dependent and stationary viewpoints. The main themes are:(1) The Mourre theory for the resolvent of self-adjoint operators(2) Two-body Schrödinger operators—Time-dependent approach and stationary approach(3) Time-dependent approach to N-body Schrödinger operators(4) Eigenfunction expansion theory for three-body Schrödinger operatorsCompared with existing books for the many-body problem, the salient feature of this book consists in the stationary scattering theory (4). The eigenfunction expansion theorem is the physical basis of Schrödinger operators. Recently, it proved to be the basis of inverse problems of quantum scattering. This book provides necessary background information to understand the physical and mathematical basis of Schrödinger operators and standard knowledge for future development.
