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Green’s Functions for First-Order Systems of Ordinary Differential Equations without the Unique Continuation Property

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Abstract

This paper is a contribution to the spectral theory associated with the differential equation \(Ju'+qu=wf\) on the real interval (a, b) when J is a constant, invertible skew-Hermitian matrix and q and w are matrices whose entries are distributions of order zero with q Hermitian and w non-negative. Under these hypotheses it may not be possible to uniquely continue a solution from one point to another, thus blunting the standard tools of spectral theory. Despite this fact we are able to describe symmetric restrictions of the maximal relation associated with \(Ju'+qu=wf\) and show the existence of Green’s functions for self-adjoint relations even if unique continuation of solutions fails.

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Notes

  1. Recall that distributions of order 0 are distributional derivatives of functions of locally bounded variation and hence may be thought of, on compact subintervals of (a, b), as measures. For simplicity we might use the word measure instead of distribution of order 0 below.

  2. A function of locally bounded variation is called balanced, if its values at any given point are averages of its left- and right-hand limits at that point.

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  1. Department of Mathematics, University of Alabama at Birmingham, Birmingham, AL, 35226-1170, USA

    Steven Redolfi & Rudi Weikard

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  1. Steven Redolfi
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  2. Rudi Weikard
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Correspondence to Rudi Weikard.

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Redolfi, S., Weikard, R. Green’s Functions for First-Order Systems of Ordinary Differential Equations without the Unique Continuation Property. Integr. Equ. Oper. Theory 94, 23 (2022). https://doi.org/10.1007/s00020-022-02703-6

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  • DOI: https://doi.org/10.1007/s00020-022-02703-6

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