Sixth-order Birkhoff regular problems

TL;DR

This paper investigates sixth-order differential equations with quadratic eigenvalue dependence, establishing conditions under which boundary value problems are Birkhoff regular, thus enabling eigenvalue asymptotics beyond self-adjoint cases.

Contribution

It provides new criteria for Birkhoff regularity in sixth-order problems with quadratic eigenvalue dependence and various boundary conditions.

Findings

Conditions for Birkhoff regularity are established.

Eigenvalue asymptotics can be derived for these problems.

Results extend regularity concepts to non-self-adjoint cases.

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 $\lambda$ is considered with classes of separable boundary conditions independent of $\lambda$ or depending linearly on $\lambda$. Conditions are given for the problems to be Birkhoff regular.