Multidimensional Periodic Schrödinger Operator: Perturbation Theories for High Energy Regions and Their Applications (Springer Tracts in Modern Physics, 291)

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
	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