JournalszaaVol. 25 , No. 3DOI 10.4171/zaa/1293

Spectral Properties of a Fourth Order Differential Equation

  • Manfred Möller

    University of Witwatersrand, Wits, South Africa
  • Vyacheslav Pivovarchik

    South-Ukrainian State Pedagogical University, Odessa, Ukraine
Spectral Properties of a Fourth Order Differential Equation cover

Abstract

The eigenvalue problem y(4)(λ,x)(gy)(λ,x)=λ2y(λ,x)y^{(4)}(\lambda,x)-(gy')'(\lambda,x)= \lambda^2y(\lambda,x) with boundary conditions y(λ,0)=0y(\lambda,0)=0, y(λ,0)=0y''(\lambda,0)=0, y(λ,a)=0y(\lambda,a)=0, y(λ,a)+iαλy(λ,a)=0y''(\lambda,a)+i \alpha\lambda y'(\lambda,a)=0 is considered, where gC1[0,a]g\in C^1[0,a] and α>0\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.