Proceedings of the Royal Society of Edinburgh: Section A Mathematics

Research Article

On Schrödinger's factorization method for Sturm-Liouville operators

U.-W. Schminckea1

a1 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 S0308210500010143_inline1 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

S0308210500010143_eqnU1

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

(Received May 05 1977)