Book contents
- Frontmatter
- Contents
- Acknowledgments
- Introduction
- 1 The KdV Hierarchy
- 2 The Combined sine-Gordon and Modified KdV Hierarchy
- 3 The AKNS Hierarchy
- 4 The Classical Massive Thirring System
- 5 The Camassa—Holm Hierarchy
- A Algebraic Curves and Their Theta Functions in a Nutshell
- B Hyperelliptic Curves of the KdV-Type
- C Hyperelliptic Curves of the AKNS-Type
- D Asymptotic Spectral Parameter Expansions and Nonlinear Recursion Relations
- E Lagrange Interpolation
- F Symmetric Functions, Trace Formulas, and Dubrovin-Type Equations
- G KdV and AKNS Darboux-Type Transformations
- H Elliptic Functions
- I Herglotz Functions
- J Spectral Measures and Weyl—Titchmarsh m-Functions for Schrödinger Operators
- List of Symbols
- Bibliography
- Index
J - Spectral Measures and Weyl—Titchmarsh m-Functions for Schrödinger Operators
Published online by Cambridge University Press: 23 December 2009
- Frontmatter
- Contents
- Acknowledgments
- Introduction
- 1 The KdV Hierarchy
- 2 The Combined sine-Gordon and Modified KdV Hierarchy
- 3 The AKNS Hierarchy
- 4 The Classical Massive Thirring System
- 5 The Camassa—Holm Hierarchy
- A Algebraic Curves and Their Theta Functions in a Nutshell
- B Hyperelliptic Curves of the KdV-Type
- C Hyperelliptic Curves of the AKNS-Type
- D Asymptotic Spectral Parameter Expansions and Nonlinear Recursion Relations
- E Lagrange Interpolation
- F Symmetric Functions, Trace Formulas, and Dubrovin-Type Equations
- G KdV and AKNS Darboux-Type Transformations
- H Elliptic Functions
- I Herglotz Functions
- J Spectral Measures and Weyl—Titchmarsh m-Functions for Schrödinger Operators
- List of Symbols
- Bibliography
- Index
Summary
Proof is the idol before whom the pure mathematician tortures himself.
Sir Arthur EddingtonIn this appendix we indicate the role of Herglotz functions(cf. Appendix I) in connection with spectral theory of Schrödinger operators on the half-line [0, ∞) and on all of ℝ. The material presented includes a discussion of (matrix-valued) spectral functions and asymptotic spectral parameter expansions of half-line Weyl–Titchmarsh m-functions for general (i.e., not necessarily Dirichlet) boundary conditions. We also treat the special case of algebro-geometric potentials and explicitly compute the corresponding spectral matrix.
We start with Schrödinger operators on the half-line [0, ∞) under the following basic assumptions.
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- Soliton Equations and their Algebro-Geometric Solutions , pp. 455 - 465Publisher: Cambridge University PressPrint publication year: 2003