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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
Affiliation:
Institut für Mathematik, RWTH Aachen

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
Copyright
Copyright © Royal Society of Edinburgh 1978

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