Spectral Properties of a Fourth Order Differential Equation

Abstract

The eigenvalue problem $y^{(4)}(\lambda ,x)-(g{y}^{\prime }{)}^{\prime }(\lambda ,x)={\lambda }^{2}y(\lambda ,x)$ with boundary conditions $y(\lambda ,0)=0$, $y^{\prime \prime }(\lambda ,0)=0$, $y(\lambda ,a)=0$, $y^{\prime \prime }(\lambda ,a)+i\alpha \lambda y^{\prime }(\lambda ,a)=0$ 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.