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Spectral Theory of Dirac Operators

Tigran Harutyunyan, Yuri Ashrafyan

Abstract

The main issues of the spectral theory of Dirac operators are presented, namely: transformation operators, asymptotics of eigenvalues and eigenfunctions, description of symmetric and self-adjoint operators in Hilbert space, expansion in eigenfunctions, uniqueness theorems in inverse problems, constructive solution of inverse problems, description of isospectral operators, and some other questions. This book is aimed at specialists in spectral theory and graduate students of mathematics at universities.

Spectral Theory of Dirac Operators

Abstract

The main issues of the spectral theory of Dirac operators are presented, namely: transformation operators, asymptotics of eigenvalues and eigenfunctions, description of symmetric and self-adjoint operators in Hilbert space, expansion in eigenfunctions, uniqueness theorems in inverse problems, constructive solution of inverse problems, description of isospectral operators, and some other questions. This book is aimed at specialists in spectral theory and graduate students of mathematics at universities.
Paper Structure (50 sections, 90 theorems, 727 equations)

This paper contains 50 sections, 90 theorems, 727 equations.

Table of Contents

  1. The Cauchy problem for the Dirac system
  2. Transformation operators
  3. Introduction and statements of the main results.
  4. The solution of equation (\ref{['c2:Q_int_eq']}).
  5. Proof of Theorem \ref{['c2:thm_2']}.
  6. The case $p, q\in L^2_{loc}$. Proof of Theorem \ref{['c2:thm_3']}.
  7. Transformation operators connected to the point $\pi$.
  8. The case $p,q\in W^{1,k}_{loc}$, $k \geq 1$.
  9. Invertibility and weak boundedness of transformation operators
  10. The Boundary value problem
  11. Existence and asymptotics of eigavalues.
  12. Asymptotics of eigenvalues in the "smooth" case.
  13. Operators in Hilbert space.
  14. The self-adjoint Dirac operators.
  15. Gradient of eigenvalue.
  16. ...and 35 more sections

Key Result

Theorem 1

If $\lambda_n(q, \frac{\pi}{2}, \frac{\pi}{2}) = n^2, \ n= 0, 1, 2, \ldots$, then $q(x) \equiv 0$.

Theorems & Definitions (145)

  • Theorem
  • Theorem 1.1
  • Lemma 1.1
  • proof
  • Lemma 1.2
  • proof
  • Lemma 1.3
  • proof
  • Theorem 1.2: Weierstrass
  • Theorem 2.1
  • ...and 135 more