Results in Mathematics

, Volume 18, Issue 1–2, pp 106–119 | Cite as

On a Conjecture of J. Weidmann

  • Frank Mantlik


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.


Spectral Function Continuous Spectrum Positive Semidefinite Boundary Behaviour Ordinary Differential Operator 
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Copyright information

© Birkhäuser Verlag, Basel 1990

Authors and Affiliations

  • Frank Mantlik
    • 1
  1. 1.Fachbereich MathematikUniversität DortmundDortmund 50Fed. Rep. of Germany

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