Elliptic boundary value problems for Bessel operators, with applications to anti-de Sitter spacetimes

TL;DR

This paper develops a rigorous mathematical framework for boundary value problems involving Bessel operators in anti-de Sitter spacetimes, establishing elliptic estimates, Fredholm properties, and eigenfunction completeness relevant to physics applications.

Contribution

It introduces a comprehensive analysis of elliptic boundary value problems for singular Bessel operators in aAdS spacetimes, including conditions for ellipticity and spectral completeness.

Findings

Established elliptic estimates near the boundary.

Proved Fredholm properties under certain conditions.

Showed completeness of eigenfunctions for Bessel operator pencils.

Abstract

This paper considers boundary value problems for a class of singular elliptic operators which appear naturally in the study of asymptotically anti-de Sitter (aAdS) spacetimes. These problems involve a singular Bessel operator acting in the normal direction. After formulating a Lopatinskii condition, elliptic estimates are established for functions supported near the boundary. The Fredholm property follows from additional hypotheses in the interior. This paper provides a rigorous framework for mode analysis on aAdS spacetimes for a wide range of boundary conditions considered in the physics literature. Completeness of eigenfunctions for some Bessel operator pencils is shown.