## The Geometry of Algebraic Fermi CurvesThe 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

1 | |

9 | |

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 |

231 | |

232 | |

Chapter 9 Monodromy for Separable Bloch Varieties | 170 |

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### Common terms and phrases

action acts assume Bloch variety branch Chapter Choose Claim Clearly complex components compute consider consists contained continuation coordinates corresponding covering critical cusp deﬁned deformation denote density derivatives determinant divisor elements equal equation fact Fermi curve ﬁber Figure ﬁrst ﬁxed points follows formula four function Furthermore germ given gives hence holomorphic implies induces intersection introduce involution irreducible isomorphism Lemma monodromy morphism multiplicity neighborhood normalization Observation obtained operator ordinary double points P1 X P1 parametrized particular path periodic polynomial potential projection Proof properties Proposition Proposition 7.2 prove Remark resp respect restriction roots simple singular singular point smooth points spectral statement strict transform subset Suppose tangent Theorem Theorem 6.1 transversal union vanishing cycle zero