Download or read book Perturbation Theory for the Schr dinger Operator with a Periodic Potential written by Yulia E. Karpeshina and published by Springer. This book was released on 2006-11-14 with total page 358 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.

Download or read book Perturbation Theory for the Schrodinger Operator with a Periodic Potential written by Yulia E. Karpeshina and published by . This book was released on 2014-01-15 with total page 364 pages. Available in PDF, EPUB and Kindle. Book excerpt:

Download or read book Multidimensional Periodic Schr dinger Operator written by Oktay Veliev and published by Springer. This book was released on 2015-03-28 with total page 249 pages. Available in PDF, EPUB and Kindle. Book excerpt: The book describes the direct problems and the inverse problem of the multidimensional Schrödinger operator with a periodic potential. This concerns perturbation theory and constructive determination of the spectral invariants and finding the periodic potential from the given Bloch eigenvalues. The unique method of this book derives the asymptotic formulas for Bloch eigenvalues and Bloch functions for arbitrary dimension. Moreover, the measure of the iso-energetic surfaces in the high energy region is construct and estimated. It implies the validity of the Bethe-Sommerfeld conjecture for arbitrary dimensions and arbitrary lattices. Using the perturbation theory constructed in this book, the spectral invariants of the multidimensional operator from the given Bloch eigenvalues are determined. Some of these invariants are explicitly expressed by the Fourier coefficients of the potential. This way the possibility to determine the potential constructively by using Bloch eigenvalues as input data is given. In the end an algorithm for the unique determination of the potential is given.

Download or read book Multidimensional Periodic Schr dinger Operator written by Oktay Veliev and published by Springer Nature. This book was released on with total page 420 pages. Available in PDF, EPUB and Kindle. Book excerpt:

Download or read book Gaps in the Dispersion Relation of the One dimensional Schr dinger Operator with Periodic Potentials written by Thomas Z. Dean and published by . This book was released on 2022 with total page 0 pages. Available in PDF, EPUB and Kindle. Book excerpt: The self-adjoint Schrödinger operator is the difference of a kinetic (Laplacian operator)and potential energy (multiplication operator). The study of this operator continues to attract the interest of many mathematicians and physicists. A commonly used mathematical approach to understand quantum mechanics is through the use of spectral and perturbation theory of the Schrödinger operator. By understanding the spectrum of the Schrödinger operator, we can understand the allowed energy states of a quantum system corresponding to a specific potential. The choice of potential dictates the behavior of the spectrum of the Schrödinger operator which in return provides insight into the behavior of the corresponding quantum system. We study periodic potentials for the Schrödinger operator because of its relation to the phenomena of Anderson localization and semi-conductor theory. A new algorithm is developed to numerically approximate the spectrum of one-dimensional periodic Schrödinger operators. From this, the behavior of spectral gaps are understood when parameters of the potential are changed (e.g. period and amplitude).Moreover, the convergence properties and the behavior of the spectrum as continuous periodic potentials are approximated by their Fourier modes are studied. The behavior of the first spectral gap for such convergences are demonstrated. These results show that the first spectral gap is well-behaved in the strong and norm resolvent convergence.

Download or read book Spectral Theory and Mathematical Physics A Festschrift in Honor of Barry Simon s 60th Birthday written by Fritz Gesztesy and published by American Mathematical Soc.. This book was released on 2007 with total page 528 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.

Download or read book Extended States for the Schr dinger Operator with Quasi Periodic Potential in Dimension Two written by Yulia Karpeshina and published by American Mathematical Soc.. This book was released on 2019-04-10 with total page 139 pages. Available in PDF, EPUB and Kindle. Book excerpt: The authors consider a Schrödinger operator H=−Δ+V(x⃗ ) in dimension two with a quasi-periodic potential V(x⃗ ). They prove that the absolutely continuous spectrum of H contains a semiaxis and there is a family of generalized eigenfunctions at every point of this semiaxis with the following properties. First, the eigenfunctions are close to plane waves ei⟨ϰ⃗ ,x⃗ ⟩ in the high energy region. Second, the isoenergetic curves in the space of momenta ϰ⃗ corresponding to these eigenfunctions have the form of slightly distorted circles with holes (Cantor type structure). A new method of multiscale analysis in the momentum space is developed to prove these results. The result is based on a previous paper on the quasiperiodic polyharmonic operator (−Δ)l+V(x⃗ ), l>1. Here the authors address technical complications arising in the case l=1. However, this text is self-contained and can be read without familiarity with the previous paper.

Download or read book Non self adjoint Schr dinger Operator with a Periodic Potential written by Oktay Veliev and published by Springer Nature. This book was released on 2021-06-19 with total page 301 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book gives a complete spectral analysis of the non-self-adjoint Schrödinger operator with a periodic complex-valued potential. Building from the investigation of the spectrum and spectral singularities and construction of the spectral expansion for the non-self-adjoint Schrödinger operator, the book features a complete spectral analysis of the Mathieu-Schrödinger operator and the Schrödinger operator with a parity-time (PT)-symmetric periodic optical potential. There currently exists no general spectral theorem for non-self-adjoint operators; the approaches in this book thus open up new possibilities for spectral analysis of some of the most important operators used in non-Hermitian quantum mechanics and optics. Featuring detailed proofs and a comprehensive treatment of the subject matter, the book is ideally suited for graduate students at the intersection of physics and mathematics.

Download or read book Mathematical Modeling in Optical Science written by Gang Bao and published by SIAM. This book was released on 2001-01-01 with total page 344 pages. Available in PDF, EPUB and Kindle. Book excerpt: This volume addresses recent developments in mathematical modeling in three areas of optical science: diffractive optics, photonic band gap structures, and waveguides. Particular emphasis is on the formulation of mathematical models and the design and analysis of new computational approaches. The book contains cutting-edge discourses on emerging technology in optics that provides significant challenges and opportunities for applied mathematicians, researchers, and engineers. Each of the three topics is presented through a series of survey papers to provide a broad overview focusing on the mathematical models. Chapters present model problems, physical principles, mathematical and computational approaches, and engineering applications corresponding to each of the three areas. Although some of the subject matter is classical, the topics presented are new and represent the latest developments in their respective fields.

Download or read book Introduction To Quantum Mechanics Schrodinger Equation And Path Integral Second Edition written by Harald J W Muller-kirsten and published by World Scientific Publishing Company. This book was released on 2012-07-19 with total page 942 pages. Available in PDF, EPUB and Kindle. Book excerpt: This text on quantum mechanics begins by covering all the main topics of an introduction to the subject. It then concentrates on newer developments. In particular it continues with the perturbative solution of the Schrödinger equation for various potentials and thereafter with the introduction and evaluation of their path integral counterparts. Considerations of the large order behavior of the perturbation expansions show that in most applications these are asymptotic expansions. The parallel consideration of path integrals requires the evaluation of these around periodic classical configurations, the fluctuation equations about which lead back to specific wave equations. The period of the classical configurations is related to temperature, and permits transitions to the thermal domain to be classified as phase transitions.In this second edition of the text important applications and numerous examples have been added. In particular, the chapter on the Coulomb potential has been extended to include an introduction to chemical bonds, the chapter on periodic potentials has been supplemented by a section on the band theory of metals and semiconductors, and in the chapter on large order behavior a section has been added illustrating the success of converging factors in the evaluation of asymptotic expansions. Detailed calculations permit the reader to follow every step.

Download or read book Floquet Theory for Partial Differential Equations written by P.A. Kuchment and published by Birkhäuser. This book was released on 2012-12-06 with total page 363 pages. Available in PDF, EPUB and Kindle. Book excerpt: Linear differential equations with periodic coefficients constitute a well developed part of the theory of ordinary differential equations [17, 94, 156, 177, 178, 272, 389]. They arise in many physical and technical applications [177, 178, 272]. A new wave of interest in this subject has been stimulated during the last two decades by the development of the inverse scattering method for integration of nonlinear differential equations. This has led to significant progress in this traditional area [27, 71, 72, 111 119, 250, 276, 277, 284, 286, 287, 312, 313, 337, 349, 354, 392, 393, 403, 404]. At the same time, many theoretical and applied problems lead to periodic partial differential equations. We can mention, for instance, quantum mechanics [14, 18, 40, 54, 60, 91, 92, 107, 123, 157-160, 192, 193, 204, 315, 367, 412, 414, 415, 417], hydrodynamics [179, 180], elasticity theory [395], the theory of guided waves [87-89, 208, 300], homogenization theory [29, 41, 348], direct and inverse scattering [175, 206, 216, 314, 388, 406-408], parametric resonance theory [122, 178], and spectral theory and spectral geometry [103 105, 381, 382, 389]. There is a sjgnificant distinction between the cases of ordinary and partial differential periodic equations. The main tool of the theory of periodic ordinary differential equations is the so-called Floquet theory [17, 94, 120, 156, 177, 267, 272, 389]. Its central result is the following theorem (sometimes called Floquet-Lyapunov theorem) [120, 267].

Download or read book Schr dinger Operators written by Hans L. Cycon and published by Springer Science & Business Media. This book was released on 1987 with total page 337 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.

Download or read book Methods of Spectral Analysis in Mathematical Physics written by Jan Janas and published by Springer Science & Business Media. This book was released on 2008-12-16 with total page 437 pages. Available in PDF, EPUB and Kindle. Book excerpt: The volume contains the proceedings of the OTAMP 2006 (Operator Theory, Analysis and Mathematical Physics) conference held at Lund University in June 2006. The conference was devoted to the methods of analysis and operator theory in modern mathematical physics. The following special sessions were organized: Spectral analysis of Schrödinger operators; Jacobi and CMV matrices and orthogonal polynomials; Quasi-periodic and random Schrödinger operators; Quantum graphs.

Download or read book Microlocal Analysis Sharp Spectral Asymptotics and Applications V written by Victor Ivrii and published by Springer Nature. This book was released on 2019-09-13 with total page 739 pages. Available in PDF, EPUB and Kindle. Book excerpt: The prime goal of this monograph, which comprises a total of five volumes, is to derive sharp spectral asymptotics for broad classes of partial differential operators using techniques from semiclassical microlocal analysis, in particular, propagation of singularities, and to subsequently use the variational estimates in “small” domains to consider domains with singularities of different kinds. In turn, the general theory (results and methods developed) is applied to the Magnetic Schrödinger operator, miscellaneous problems, and multiparticle quantum theory. In this volume the methods developed in Volumes I, II, III and IV are applied to multiparticle quantum theory (asymptotics of the ground state energy and related problems), and to miscellaneous spectral problems.

Download or read book Boundary Value Problems of Mathematical Physics written by Olʹga A. Ladyženskaja and published by American Mathematical Soc.. This book was released on 1991 with total page 254 pages. Available in PDF, EPUB and Kindle. Book excerpt:

Download or read book Perturbation Theory in Periodic Problems for Two Dimensional Integrable Systems written by I. M. Krichever and published by CRC Press. This book was released on 1992 with total page 118 pages. Available in PDF, EPUB and Kindle. Book excerpt:

Download or read book Schr dinger Operators written by Hans L. Cycon and published by Springer. This book was released on 2009-08-19 with total page 337 pages. Available in PDF, EPUB and Kindle. Book excerpt: A complete understanding of Schrödinger operators is a necessary prerequisite for unveiling the physics of nonrelativistic quanturn mechanics. Furthermore recent research shows that it also helps to deepen our insight into global differential geometry. This monograph written for both graduate students and researchers summarizes and synthesizes the theory of Schrödinger operators emphasizing the progress made in the last decade by Lieb, Enss, Witten and others. Besides general properties, the book covers, in particular, multiparticle quantum mechanics including bound states of Coulomb systems and scattering theory, quantum mechanics in constant electric and magnetic fields, Schrödinger operators with random and almost periodic potentials and, finally, Schrödinger operator methods in differential geometry to prove the Morse inequalities and the index theorem.