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Perturbation Theory for the Schrödinger Operator with a Periodic Potential

2006-11-14 Mathematics
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Release : 2006-11-14
Genre : Mathematics
Kind : eBook
Book Rating : 561/5 ( reviews)

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Book Synopsis Perturbation Theory for the Schrödinger Operator with a Periodic Potential by : Yulia E. Karpeshina

Download or read book Perturbation Theory for the Schrödinger Operator with a Periodic Potential written by Yulia E. Karpeshina. This book was released on 2006-11-14. 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.

Perturbation Theory for the Schrodinger Operator with a Periodic Potential

2014-01-15
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Release : 2014-01-15
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Kind : eBook
Book Rating : 660/5 ( reviews)

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Book Synopsis Perturbation Theory for the Schrodinger Operator with a Periodic Potential by : Yulia E. Karpeshina

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

Multidimensional Periodic Schrödinger Operator

2015-03-28 Science
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Release : 2015-03-28
Genre : Science
Kind : eBook
Book Rating : 433/5 ( reviews)

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Book Synopsis Multidimensional Periodic Schrödinger Operator by : Oktay Veliev

Download or read book Multidimensional Periodic Schrödinger Operator written by Oktay Veliev. This book was released on 2015-03-28. 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.

Multidimensional Periodic Schrödinger Operator

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Book Rating : 355/5 ( reviews)

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Book Synopsis Multidimensional Periodic Schrödinger Operator by : Oktay Veliev

Download or read book Multidimensional Periodic Schrödinger Operator written by Oktay Veliev. This book was released on . Available in PDF, EPUB and Kindle. Book excerpt:

Gaps in the Dispersion Relation of the One-dimensional Schrödinger Operator with Periodic Potentials

2022 Schrödinger operator
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Release : 2022
Genre : Schrödinger operator
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Book Synopsis Gaps in the Dispersion Relation of the One-dimensional Schrödinger Operator with Periodic Potentials by : Thomas Z. Dean

Download or read book Gaps in the Dispersion Relation of the One-dimensional Schrödinger Operator with Periodic Potentials written by Thomas Z. Dean. This book was released on 2022. 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.

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