Spectral Properties of a Fourth Order Differential Equation

Abstract

The eigenvalue problem $y^{(4)}(\lambda,x)-(gy')'(\lambda,x)= \lambda^2y(\lambda,x)$ with boundary conditions $y(\lambda,0)=0$, $y''(\lambda,0)=0$, $y(\lambda,a)=0$, $y''(\lambda,a)+i \alpha\lambda y'(\lambda,a)=0$ is considered, where $g\in C^1[0,a]$ and $\alpha >0$. It is shown that the eigenvalues lie in the closed upper half-plane and on the negative imaginary axis. A formula for the asymptotic distribution of the eigenvalues is given and the location of the pure imaginary spectrum is investigated.