## 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

0)-vanishing cycles algebraic analytic continuation arithmetic genus Aspec assume B(q)comp band function Bloch variety branch Chapter coefﬁcient compactiﬁcation components compute consists coordinates Corollary curve F cusp vanishing cycle deﬁned Deﬁnition deformation denote density determinant divisor eigenvalues equation Euler characteristic Fermi curve ﬁber Figure ﬁnd ﬁnite ﬁrst ﬁxed points fulﬁlled Furthermore germ holomorphic Hove vanishing cycle hyperelliptic curves implies independent electron approximation inﬁnity involution irreducible isomorphism matrix monodromy group morphism neighborhood Observation ordinary double points P1 X P1 parallel transport parametrized path Picard-Lefschetz formula point of B(O point q polynomial potential proof of Lemma proof of Proposition proof of Theorem Proposition 7.2 prove quadrics ramiﬁcation Remark resp Sing(B singular point smooth points spectral quadruple point spectral van Hove spectral vanishing cycle strict transform sufﬁciently small tacnode tacnode point Theorem 6.1 transversal V E L2 van Hove singularities Zariski open zero