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On Schrödinger's factorization method for Sturm-Liouville operators

Published online by Cambridge University Press:  14 November 2011

U.-W. Schmincke
U.-W. Schmincke
Affiliation:
Institut für Mathematik, RWTH Aachen
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Synopsis

We consider the Friedrichs extension A of a minimal Sturm-Liouville operator L0 and show that A admits a Schrödinger factorization, i.e. that one can find first order differential operators Bk with where the μk are suitable numbers which optimally chosen are just the lower eigenvalues of A (if any exist). With the help of this theorem we derive for the special case L0u = −u″ + q(x)u with q(x) → 0 (|x| → ∞) the inequality

σd(A) being the discrete spectrum of A. This inequality is seen to be sharp to some extent.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 80 , Issue 1-2 , 1978 , pp. 67 - 84
Copyright
Copyright © Royal Society of Edinburgh 1978

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