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Expansions in terms of sets of functions with complex eigenvalues

Published online by Cambridge University Press:  24 October 2008

R. Peierls
R. Peierls
Affiliation:
The UniversityBirmingham
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Extract

In the following I discuss the properties, in particular the completeness of the set of eigenfunctions, of an eigenvalue problem which differs from the well-known Sturm-Liouville problem by the boundary condition being of a rather unusual type.

The problem arises in the theory of nuclear collisions, and for our present purpose we take it in the simplified form

where 0 ≤ x ≤ 1. V(x) is a given real function, which we assume to be integrable and to remain between the bounds ± M, and W is an eigenvalue. The eigenfunction ψ(x) is subject to the boundary conditions

and

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 44 , Issue 2 , April 1948 , pp. 242 - 250
Copyright
Copyright © Cambridge Philosophical Society 1948

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References

† Kapur, P. L. and Peierls, R., Proc. Roy. Soc. A, 166 (1938), 277.CrossRefGoogle Scholar

† These restrictions are imposed on F for simplicity. It is probable that the results would equally apply to any function that can be expanded into a Fourier series.