On square root domains for non-self-adjoint Sturm-Liouville operators

TL;DR

This paper characterizes the square root domains of non-self-adjoint Sturm-Liouville operators with general coefficients and boundary conditions across various domains, extending understanding of their functional calculus.

Contribution

It provides a comprehensive analysis of square root domains for a broad class of non-self-adjoint Sturm-Liouville operators with general boundary conditions.

Findings

Explicit descriptions of square root domains for different boundary conditions.

Extension of results to operators on the entire real line, half-line, and bounded intervals.

Handling of general coefficients under minimal regularity assumptions.

Abstract

We determine square root domains for non-self-adjoint Sturm--Liouville operators of the type $$ L_{p,q,r,s} = - \frac{d}{dx}p\frac{d}{dx}+r\frac{d}{dx}-\frac{d}{dx}s+q $$ in $L^2((c,d);dx)$, where either $(c,d)$ coincides with the real line $\mathbb{R}$, the half-line $(a,\infty)$, $a \in \mathbb{R}$, or with the bounded interval $(a,b) \subset \mathbb{R}$, under very general conditions on the coefficients $q, r, s$. We treat Dirichlet and Neumann boundary conditions at $a$ in the half-line case, and Dirichlet and/or Neumann boundary conditions at $a,b$ in the final interval context. (In the particular case $p=1$ a.e.\ on $(a,b)$, we treat all separated boundary conditions at $a, b$.)