Degenerate operators on the half-line

TL;DR

This paper investigates elliptic and parabolic differential equations on the half-line involving singular elliptic operators with a potential, analyzing their properties and solutions.

Contribution

It introduces a comprehensive analysis of elliptic and parabolic problems governed by a class of singular elliptic operators with potential functions on the half-line.

Findings

Characterization of the operators' spectral properties

Existence and uniqueness results for solutions

Regularity and asymptotic behavior of solutions

Abstract

We study elliptic and parabolic problems governed by the singular elliptic operators $$ y^{\alpha}\left(D_{yy}+\frac{c}{y}D_y\right)-V(y),\qquad\alpha \in\mathbb R $$ in $\mathbb R_+$, where $V$ is a potential having non-negative real part.