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Journal of the London Mathematical Society 1999 59(1):188-206; doi:10.1112/S0024610799007012
© 1999 by London Mathematical Society
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© The London Mathematical Society

Spectral Analysis of Higher-Order Differential Operators II: Fourth-Order Equations

Christian Remling

FB Mathematik/Informatik, Universität Osnabrück 49069 Osnabrück, Germany. E-mail: christian.remling{at}mathematik.uni-osnabrueck.de

Received 21 May 1996. Revision received 9 September 1996.

We investigate the location and nature of the spectrum of the fourth-order self-adjoint equation

(p0 y'')''+(p1 y')'+qy=zwy

subject to certain asymptotic assumptions on the coefficients. The main tools are the theory of asymptotic integration and the Titchmarsh–Weyl M-matrix. Asymptotic integration yields asymptotic formulae for the solutions of the differential equation which are then used to derive properties of the M-matrix. The characterisation of spectral properties in terms of the boundary behaviour of M leads to the desired results.


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