UID:
edoccha_9961426837502883
Format:
1 online resource (420 pages)
Edition:
Third edition.
ISBN:
9783031490354
Series Statement:
Springer Tracts in Modern Physics Series ; Volume 291
Note:
Intro -- Preface -- Contents -- 1 Preliminary Facts -- 1.1 Lattices, Brillouin Zones, and Periodic Functions in double struck upper R Superscript dmathbbRd -- 1.2 Schrödinger Operator, Bloch Functions -- 1.3 Band Structure, Fermi Surfaces, and Perturbations -- 1.4 Some Discussions of the Perturbation Theory -- References -- 2 From One-Dimensional to Multidimensional -- 2.1 One-Dimensional Schrödinger Operator and Comparisons with Multidimensional Ones -- 2.1.1 Introduction and Some Discussions -- 2.1.2 On the One-Dimensional Schrödinger Operator with a Periodic Potential -- 2.1.3 One-Dimensional Schrödinger Operator with a Matrix Potential -- 2.2 Asymptotic Formulas for Eigenvalues in Two- and Three-Dimensional Cases -- 2.2.1 On the Iterations of (2.1.10) -- 2.2.2 Asymptotic Formulas for the Non-resonance Eigenvalues -- 2.2.3 Single Resonance Eigenvalues and Matrices -- 2.2.4 Estimations of the Resonance and Non-resonance Sets -- 2.3 Simple Sets and Bloch Functions in Dimensions Two and Three -- 2.3.1 Discussion of the Simplicity and the Asymptotic Formulas for the Bloch Functions -- 2.3.2 Precise Construction of the Simple Set in the Dimension Two -- 2.3.3 Precise Construction of the Simple Set in the Dimension Three -- 2.4 Estimations of the Simple Sets and Isoenergetic Surfaces -- 2.4.1 Some Properties of the Sets (2.4.7) and Their Applications -- 2.4.2 The Proof of the Main Theorems -- 2.4.3 The Proofs of the Estimations (2.4.18), (2.4.21), (2.4.23), and (2.4.24) -- 2.5 On the Nonsmooth Potentials -- 2.5.1 Bloch Eigenvalues for the Nonsmooth Potentials -- 2.5.2 Bloch Functions for the Nonsmooth Potentials -- 2.5.3 Estimations of the Simple Sets for the Nonsmooth Potentials -- References -- 3 Asymptotic Formulas for the Bloch Eigenvalues and Bloch Functions -- 3.1 Introduction -- 3.2 Asymptotic Formulae for the Eigenvalues.
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3.3 Bloch Eigenvalues Near the Diffraction Planes -- 3.4 Asymptotic Formulas for the Bloch Functions -- 3.5 Simple Sets and Isoenergetic Surfaces -- 3.6 Bloch Functions Near the Diffraction Hyperplanes -- References -- 4 Constructive Determination of the Spectral Invariants -- 4.1 Introduction and Preliminary Facts -- 4.2 First and Second Terms of the Asymptotics -- 4.3 On the Derivatives of the Band Functions -- 4.4 The Construction of the Spectral Invariants -- 4.5 Appendices -- References -- 5 Periodic Potential From the Spectral Invariants -- 5.1 Introduction -- 5.2 On the Simple Invariants -- 5.3 Finding the Fourier Coefficients Corresponding to the Boundary -- 5.4 Inverse Problem in a Dense Set -- 5.5 Finding the Simple Potential From the Invariants -- 5.6 On the Stability of the Algorithm -- 5.7 Uniqueness Theorems -- References -- 6 Conclusions and Some Generalization -- 6.1 Conclusions -- 6.2 On Some Generalizations -- 6.2.1 Asymptotic Formulas for the Operators From upper S upper B upper C left parenthesis upper H right parenthesisSBC(H) -- 6.2.2 On the Continuity of the Eigenvalues and Eigenfunctions -- References -- Appendix Index -- Index.
Additional Edition:
ISBN 9783031490347
Language:
English
