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A Brief Explanation of the Spectral Expansion Method for Non-Self-Adjoint Differential Operators with Periodic Coefficients

O. A. Veliev

Abstract

In this paper, we briefly explain the spectral expansion problem for differential operators defined on the entire real line, generated by a differential expression with periodic, complex-valued coefficients.

A Brief Explanation of the Spectral Expansion Method for Non-Self-Adjoint Differential Operators with Periodic Coefficients

Abstract

In this paper, we briefly explain the spectral expansion problem for differential operators defined on the entire real line, generated by a differential expression with periodic, complex-valued coefficients.

Paper Structure

This paper contains 2 theorems, 38 equations.

Key Result

Theorem 1

We have the elegant spectral decompositions if and only if $L(q)$ has no ESS and ESS at infinity, where for $\lambda\in\sigma(L_{t}(q)),$ $F(\lambda)=\varphi^{\prime}(1,\lambda)+\theta(1,\lambda),$$\varphi (x,\lambda)$ and $\theta(x,\lambda)$ are the solutions of satisfying the conditions $\varphi(0,\lambda)=0,$$\varphi^{\prime}(0,\lambda)=1$ and $\theta(0,\lambda)=1,$$\theta^{\prime}(0,\lambda

Theorems & Definitions (6)

  • Example 1
  • Example 2
  • Definition 1
  • Definition 2
  • Theorem 1
  • Theorem 2