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Eigenfunction expansions and spectral matrices of singular differential operators

Published online by Cambridge University Press:  14 November 2011

Don B. Hinton
Don B. Hinton
Affiliation:
Mathematics Department, University of Tennessee, Knoxville, Tennessee, U.S.A.
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Synopsis

We consider the eigenfunction expansions associated with a symmetric differential operator M[·] of order 2n with coefficients defined on an open interval (a, b). Each singular endpoint of (a, b) is assumed to be of limit-n type. A direct convergence theory is established for the eigenfunction series expansion of a function y in a set Termwise differentiation of the series is established for the derivatives of order up to n. For O ≤ i ≤ n − 1, the i-fold differentiated series converges absolutely and uniformly to y(i) on compact intervals; the n−fold differentiated series converges to yn in the mean. The expansion theory is valid also when an essential spectrum is present. An explicit formula is given for the calculation of the spectral matrix.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 80 , Issue 3-4 , 1978 , pp. 289 - 308
Copyright
Copyright © Royal Society of Edinburgh 1978

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