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On a Conjecture of J. Weidmann

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Abstract

Consider the Sturm-Liouville differential expression l(y) = −y″ +q(x)y on an interval (a,b) and assume that l is in the limit point case at b. Fix c ∈(a,b) and let L, Lb be self-adjoint realizations of l in ℒ2(a,b), ℒ2(c,b) respectively. If Lb has purely absolutely continuous spectrum in an interval J and if the spectral function ρb of Lb satisfies some mild growth conditions then the spectrum of L in J is shown to be purely absolutely continuous, too. Our result confirms a conjecture of J. Weidmann (1982). It had been shown by del Rio Castillo (1988) that in Weidmann’s original formulation this conjecture is false.

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  1. Fachbereich Mathematik, Universität Dortmund, Postfach 50 05 00, D-4600, Dortmund 50, Fed. Rep. of Germany

    Frank Mantlik

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  1. Frank Mantlik
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Mantlik, F. On a Conjecture of J. Weidmann. Results. Math. 18, 106–119 (1990). https://doi.org/10.1007/BF03323158

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