Liouville-Riemann-Roch Theorems on Abelian Coverings

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Preface Acknowledgements Contents 1 Preliminaries 1.1 Periodic Elliptic Operators on Abelian Coverings 1.2 Floquet Transform 1.3 Bloch and Fermi Varieties 1.4 Floquet-Bloch Functions and Solutions 1.5 Liouville Theorem on Abelian Coverings 1.6 Some Properties of Spaces VpN(A) 1.7 Explicit Formulas for Dimensions of Spaces V∞N(A) 1.8 The Nadirashvili-Gromov-Shubin Version of the Riemann-Roch Theorem for Elliptic Operators on Noncompact Manifolds 1.8.1 Some Notions and Preliminaries 1.8.2 Point Divisors 1.8.3 Rigged Divisors 1.8.4 Nadirashvili-Gromov-Shubin Theorem on Noncompact Manifolds 2 The Main Results 2.1 Non-empty Fermi Surface 2.1.1 Assumptions 2.1.2 Spaces 2.1.3 Results 2.2 Empty Fermi Surface 3 Proofs of the Main Results 3.1 Some Notions 3.2 Proof of Theorem 2.2 3.3 Proof of Theorem 2.5 3.3.1 Proof of Proposition 2.7 3.4 Proof of Theorem 2.8 3.4.1 Proof of Proposition 2.9 3.4.2 Proof of Proposition 2.10 3.4.3 Proof of Proposition 2.11 3.4.4 Proof of Corollary 2.12 3.5 Proof of Theorem 2.16 4 Specific Examples of Liouville-Riemann-Roch Theorems 4.1 Self-Adjoint Operators 4.1.1 Periodic Operators with Non-degenerate Spectral Edges 4.1.2 Periodic Operators with Dirac Points 4.2 Non-Self-Adjoint Second Order Elliptic Operators 5 Auxiliary Statements and Proofs of Technical Lemmas 5.1 Properties of Floquet Functions on Abelian Coverings 5.2 Basic Properties of the Family {A(k)}k Cd 5.3 Properties of Floquet Transforms on Abelian Coverings 5.4 A Schauder Type Estimate 5.5 A Variant of Dedekind's Lemma 5.6 Proofs of Some Other Technical Statements 5.6.1 Proof of Theorem 1.13 5.6.2 Proof of Theorem 1.14 5.6.3 Proof of Corollary 1.27 A Final Remarks and Conclusions References Index