On the Spectrality of the Differential Operators with Periodic Coefficients

TL;DR

This paper addresses when differential operators with periodic coefficients on $L_2(\mathbb{R})$ are spectral operators in the Dunford sense. By combining the asymptotic spectrality framework from prior work with Bloch-eigenvalue localization, the authors derive explicit, quantitative criteria that guarantee full spectrality for both odd and even order cases: (i) for odd $n$, a smallness condition on the higher-coefficient norm $C$ ensures spectrality and yields a Dunford spectral expansion $f(x)=\frac{1}{2}\sum_{k\in\mathbb{Z}}\int_{-1}^1 a_k(t) \Psi_{k,t}(x)\,dt$; (ii) for even $n>2$, a real constant $c$ with $p_1(x)=c$ and a corresponding bound guarantees spectrality, with a parallel $n=2$ case $|c|>\tfrac{1}{2}\|q\|$. The results clarify when such non-self-adjoint periodic operators admit uniformly bounded spectral projections and provide explicit spectral expansions, while also highlighting that spectrality is rare in the even-order setting without extra structure.

Abstract

In this paper, we establish a condition on the coefficients of differential operators generated in the space of square-integrable functions on the entire real line by an ordinary differential expression with periodic, complex-valued coefficients, under which the operator is a spectral operator in the sense of Dunford [1].

Theorems & Definitions (9)

• Example 1
• Example 2
• Example 3
• Theorem 1
• Theorem 2
• Lemma 1
• Lemma 2
• Theorem 3
• Corollary 1