Canonical forms for boundary conditions of self-adjoint odd-order differential operators

TL;DR

This paper derives canonical boundary condition forms for self-adjoint odd-order differential operators, extending previous work on even-order cases, to facilitate spectral analysis and numerical applications.

Contribution

It introduces canonical forms for boundary conditions of self-adjoint odd-order differential operators with eigenvalue dependence, filling a gap in the existing theory.

Findings

Provides explicit canonical forms for boundary conditions of odd-order operators

Extends previous even-order canonical form results to odd-order cases

Facilitates spectral analysis and numerical methods for these operators

Abstract

It is useful to have canonical forms of boundary conditions in the study of the eigenvalues of boundary value problems and associated numerical applications. In [J. Appl. Anal. Comput., 2024, 14(4), {1854--1868}], a canonical form is given for self-adjoint differential operators of even order, with eigenvalue parameter dependent boundary conditions. In this article, we derive canonical forms for the remaining case, namely: for self-adjoint $(2n+1)$-th order differential operators with eigenvalue parameter dependent boundary conditions.