# Boundary Conditions associated with the General Left-Definite Theory for Differential Operators

## TL;DR

This paper characterizes the domains of self-adjoint differential operators in the left-definite theory using classical boundary conditions, bridging abstract spectral theory with explicit boundary condition descriptions.

## Contribution

It provides explicit boundary condition characterizations for operators with orthogonal eigenfunctions within the left-definite framework, extending prior abstract domain descriptions.

## Key findings

- Domains characterized by classical boundary conditions
- Applicable to operators with orthogonal eigenfunctions
- Bridges abstract theory with explicit boundary conditions

## Abstract

In the early 2000's, Littlejohn and Wellman developed a general left-definite theory for certain self-adjoint operators by fully determining their domains and spectral properties. The description of these domains do not feature explicit boundary conditions. We present characterizations of these domains given by the left-definite theory for all operators which possess a complete system of orthogonal eigenfunctions, in terms of classical boundary conditions.