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Zeitschrift für Analysis und ihre Anwendungen

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Volume 25, Issue 3, 2006, pp. 341–366
DOI: 10.4171/ZAA/1293

Published online: 2006-09-30

Spectral Properties of a Fourth Order Differential Equation

Manfred Möller[1] and Vyacheslav Pivovarchik[2]

(1) University of Witwatersrand, Wits, South Africa
(2) South-Ukrainian State Pedagogical University, Odessa, Ukraine

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.

Keywords: Fourth-order differential equation, pure imaginary eigenvalues, eigenvalue distribution

Möller Manfred, Pivovarchik Vyacheslav: Spectral Properties of a Fourth Order Differential Equation. Z. Anal. Anwend. 25 (2006), 341-366. doi: 10.4171/ZAA/1293