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The least limit point of the spectrum associated with sinǵular differential operators

Published online by Cambridge University Press:  24 October 2008

M. S. P. Eastham
M. S. P. Eastham
Affiliation:
The University, Southampton
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Extract

Let τ be the formally self-adjoint differential operator denned by

where the pr(x) are real-valued, , and p0(x) > 0. Then τ determines a real symmetric linear operator T0, given by T0f = τf, whose domain D(T0) consists of those functions f in the complex space L2(0, ∞) which have compact support and 2n continuous derivatives in (0, ∞) and vanish in some right neighbourhood of x = 0 ((7), p. 27–8). Since D(T0) is dense in L2(0, ∞), T0 has a self-adjoint extension T. We denote by μ the least limit point of the spectrum of T. The operator T may not be unique, but all such T have the same essential spectrum ((7), p. 28) and therefore μ does not depend on the choice of T.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 67 , Issue 2 , March 1970 , pp. 277 - 281
Copyright
Copyright © Cambridge Philosophical Society 1970

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References

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