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Families of Spectral Measures with Mixed Types

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Book cover Operator Methods in Ordinary and Partial Differential Equations

Part of the book series: Operator Theory: Advances and Applications ((OT,volume 132))

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Abstract

Consider a family of Sturm-Liouville operators H θ on the half-axis defined as

$${{H}_{\theta }}u = - u'' + q(x)u 0 \leqslant x < \infty $$

with the boundary condition

$$ u(0)\cos \theta + u'(0)\sin \theta = 0 0 \leqslant \theta < \pi $$

and the limit point case at infinity. We show that it is possible for all H θ to have dense absolutely continuous and dense singular spectrum. The construction is based on integral representations of Pick functions in the upper half-plane. We also discuss applications to the Krein spectral shift.

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Author information

Authors and Affiliations

  1. IIMAS-UNAM Circuito Escolar, Ciudad Universitaria, 04510, D.F., México

    R. del Rio

  2. Cajamarca 104 Col. Las Americas Naucalpan de Juarez, Edo. de Mexico, 53040, Mexico

    S. Fuentes

  3. Department of Mathematics, College Station, Texas, 77843-3368, USA

    A. Poltoratski

Authors
  1. R. del Rio
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  2. S. Fuentes
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  3. A. Poltoratski
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Editor information

Editors and Affiliations

  1. Institute of Applied Mathematics, University of Bonn, Bonn, 53115, Germany

    Sergio Albeverio

  2. Department of Physics, Stockholm University, Stockholm, 10691, Sweden

    Nils Elander

  3. School of Mathematics and Statistics, University of Birmingham, Edgbaston, Birmingham, England, B15 2TT, UK

    W. Norrie Everitt

  4. Department of Mathematics, Stockholm University, Stockholm, 10691, Sweden

    Pavel Kurasov

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© 2002 Springer Basel AG

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del Rio, R., Fuentes, S., Poltoratski, A. (2002). Families of Spectral Measures with Mixed Types. In: Albeverio, S., Elander, N., Everitt, W.N., Kurasov, P. (eds) Operator Methods in Ordinary and Partial Differential Equations. Operator Theory: Advances and Applications, vol 132. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-8219-4_12

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  • DOI: https://doi.org/10.1007/978-3-0348-8219-4_12

  • Publisher Name: Birkhäuser, Basel

  • Print ISBN: 978-3-0348-9479-1

  • Online ISBN: 978-3-0348-8219-4

  • eBook Packages: Springer Book Archive

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