The Geometry of Algebraic Fermi Curves

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Academic Press, Dec 2, 2012 - Mathematics - 246 pages
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The Geometry of Algebraic Fermi Curves deals with the geometry of algebraic Fermi curves, with emphasis on the inverse spectral problem. Topics covered include the periodic Schrödinger operator and electrons in a crystal; one-dimensional algebraic Bloch varieties; separable Bloch varieties; and monodromy for separable and generic Bloch varieties. Compactification, the potential zero, and density of states are also discussed.
This book consists of 13 chapters and begins by recalling the static lattice approximation for electronic motion at low temperature in a pure, finite sample of a d-dimensional crystal. The position of the Fermi energy and the geometry of the Fermi hypersurface in relation to the metallic properties of the crystal are described. The following chapters focus on the Bloch variety associated with a discrete two-dimensional periodic Schrödinger operator; algebraic Bloch varieties in one dimension; compactification of the Bloch variety; and the potential zero. The geometry of the Bloch variety of a separable potential is also considered, along with the topology of the family of Fermi curves. The final chapter demonstrates how the Bloch variety is determined by the density of states.
This monograph will be a useful resource for students and teachers of mathematics.
 

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Contents

Chapter 1 The Periodic Schrödinger Operator and Electrons in a Crystal
1
Chapter 2 Preliminaries
9
Chapter 3 OneDimensional Algebraic Bloch Varieties
15
Chapter 4 Compactification and Consequences
39
Chapter 5 The Potential Zero
61
Chapter 6 Separable Bloch Varieties
101
Chapter 7 Topology of the Family of Fermi Curves
125
Chapter 8 Monodromy
163
Chapter 10 Monodromy for Generic Bloch Varieties
188
Chapter 11 Density of States
205
Chapter 12 Density of States and Monodromy
214
Chapter 13 The Density of States Determines the Bloch Variety
222
References
228
Index of Notation
231
Index
232
Copyright

Chapter 9 Monodromy for Separable Bloch Varieties
170

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