Asymptotic Distribution of Eigenvalues of Differential Operators

ISBN-13: 9780792305392 / Angielski / Twarda / 1990 / 280 str.

Serge Levendorskii

Opis

One service mathematics has rendered the 'Et moi, ..., si j'avait su comment en revenir, je n'y serais point alle.' human race. It has put common sense back Jules Verne where it belongs, on the topmost shelf next to the dusty canister labelled 'discarded n- sense'. The series is divergent; therefore we may be able to do something with it. Eric T. Bell O. Heaviside Mathematics is a tool for thought. A highly necessary tool in a world where both feedback and non linearities abound. Similarly, all kinds of parts of mathematics serve as tools for other parts and for other sciences. Applying a simple rewriting rule to the quote on the right above one finds such statements as: 'One service topology has rendered mathematical physics .. .'; 'One service logic has rendered com puter science .. .'; 'One service category theory has rendered mathematics .. .'. All arguably true. And all statements obtainable this way form part of the raison d'etre of this series.

Spis treści

1. The Weyl-Hörmander Calculus of Pseudodifferential Operators.- §1. Classes of Symbols.- §2. Estimates for Solutions of Schrödinger-Type Equations.- §3. The Fundamental Theorems of Calculus.- §4. Continuity of Pseudodifferential Operators.- §5. Weight Spaces of Sobolev Type.- §6. Action of Pseudodifferential Operators in Weight Spaces.- 2. Basic Theorems of the Method of Approximate Spectral Projection for Scalar and Matrix Operators.- §7. Formulation of the Basic Theorems.- §8. Auxiliary Propositions.- §9. Proof of Theorem 7.2 for the Scalar Case.- §10. Proof of Theorem 7.3 for the Scalar Case.- §11. Proofs of Theorems 7.2 and 7.3 for the Matrix Case.- §12. Proofs of Theorems 7.1, 7.4, and 7.5.- 3. Operators in a Bounded Domain.- §13. Douglis-Nirenberg Elliptic Operators. Dirichlet-Type Problems.- §14. General Boundary Value Problems for Elliptic Operators.- §15. Problems with Resolvable Constraints.- §16. Electromagnetic Resonator.- §17. Asymptotics of the Discrete Spectrum of Douglis—Nirenberg Operators with a Totally Disconnected Essential Spectrum.- §18. Linearized Stationary Navier—Stokes System.- §19. Asymptotics for Eigenfrequencies of a Shell in a Vacuum.- 4. Operators in Unbounded Domains.- §20. Schrödinger Operators with Increasing Potential.- §21. Asymptotics of a Discrete Spectrum of Schrödinger Operators and Dirac Operators with Decreasing Potentials.- 5. Asymptotics of the Spectrum of Pseudodifferential Operators with Operator-Valued Symbols and Some Applications.- §22. Pseudodifferential Operators with Operator-Valued Symbols.- §23. Boundary Value Problems in Strongly Anisotropic Domains.- 6. Degenerate Differential Operators.- §24. General Analysis of Degenerate Operators and Generalizations of the Weyl Formula.- §25. Schrödinger Operators with Degenerate Homogeneous Potential.- §26. Model Problems for Degenerate Differential Operators in a Bounded Domain.- §27. Degenerate Differential Operators in a Bounded Domain.- §28. Degenerate Differential Operators in an Unbounded Domain.- Appendix: Basic Variational Theorems.- A Brief Review of the Bibliography.- Notation Index.