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Using Root Functions for Eigenvalue Problems of Ordinary Differential Operators
Authors:Oksana Guba
Institution:1. Department of Mathematics and Statistics , University of New Mexico , Albuquerque, New Mexico, USA Oksana@unm.edu
Abstract:We study eigenvalue problems for an ordinary differential operator L acting on L 2(?)-spaces (Problem 1) and on L 2(J)-spaces (Problem 2). Here J is a bounded but large interval. Assuming that in Problem 1 the spectral parameter s lies in the set of normal points of L, we show that the structure of eigenspaces for both problems is similar to the structure of finite complex-valued matrices. In the case of a finite matrix, the geometry of eigenspaces is described by the Jordan form. In the case of ordinary differential operators, the corresponding geometry is described by a sequence of root functions. Therefore, the main tool of our studies is root functions for complex-valued analytical matrix functions.
Keywords:Boundary conditions  Boundary value problem  Root functions  Spectra of ordinary differential operators
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