Sixth-order Birkhoff regular problems

TL;DR

This work extends Birkhoff regularity to sixth-order differential problems with operator pencils $L(λ)=λ^{2}M-iλK-A$ that may be non-self-adjoint. It develops a determinant-based framework using Fourier-transform techniques to analyze boundary-condition matrices and classifies boundary-condition exponents, establishing that regularity holds exactly when there exist endpoint indices $r_0,r_1$ satisfying the conditions $C(r_0,u)$ and $C(r_1,u)$. The results provide a rigorous, case-rich classification of when sixth-order problems admit Birkhoff-type eigenfunction expansions, setting the stage for asymptotic eigenvalue analysis in forthcoming work. The methods unify high-order regularity criteria with explicit boundary-parameter constraints, enabling future spectral-characterization studies.

Abstract

Asymptotics of the eigenvalues can always be derived for self-adjoint boundary value problems. However, they can also be derived for boundary value problems that fail to be self-adjoint provided that they are Birkhoff regular. A regular sixth-order differential equation that depends quadratically on the eigenvalue parameter $λ$ is considered with classes of separable boundary conditions independent of $λ$ or depending linearly on $λ$. Conditions are given for the problems to be Birkhoff regular.

Theorems & Definitions (15)

• Lemma 1
• Proposition 1
• Proposition 2
• Proposition 3
• Proposition 4
• Lemma 2
• Lemma 3
• Lemma 4
• Lemma 5
• Lemma 6