Using Root Functions for Eigenvalue Problems of Ordinary Differential Operators

Abstract

We study eigenvalue problems for an ordinary differential operator L acting on L ₂(ℝ)-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

AMS Subject Classification:

• 34L05
• 34L10
• 34B09
• 34B40

ACKNOWLEDGMENT

The author wishes to thank Professor Pedro Embid and the reviewers for their comments, which significantly improved the manuscript.