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A class of nonselfadjoint spectral differential operators of interest in physics

Victor Laliena

Abstract

It is shown that the nonselfadjoint (and non-normal) linear ordinary differential operators of a certain class are spectral operators of scalar type in the sense of Dunford and Bade. Operators of this kind appear in physical problems such as the scattering of spin waves by magnetic solitons.

A class of nonselfadjoint spectral differential operators of interest in physics

Abstract

It is shown that the nonselfadjoint (and non-normal) linear ordinary differential operators of a certain class are spectral operators of scalar type in the sense of Dunford and Bade. Operators of this kind appear in physical problems such as the scattering of spin waves by magnetic solitons.

Paper Structure

This paper contains 11 sections, 24 theorems, 129 equations.

Table of Contents

  1. Introduction
  2. Summary of main results
  3. Spectrum of ${L}$ and ${L^{*}}$
  4. Spectral projections for bounded Borel sets
  5. The Green function of ${L}$
  6. Spectral expansion
  7. The bounded linear maps $U_n$ and $V_n$
  8. Spectral resolution of the identity for ${L}$
  9. Final remarks
  10. Proof of theorem \ref{['thm:sols_edo']}
  11. Proof of an ancillary result

Key Result

Theorem 1

$\rho({L})=\rho({L^{*}})=\rho({\Lambda})$, and if $z\in\rho({L})$ then

Theorems & Definitions (47)

  • Theorem 1
  • proof
  • Theorem 2
  • proof
  • Theorem 3
  • proof
  • Theorem 4
  • proof
  • Lemma 5
  • proof
  • ...and 37 more