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Transmutations and Spectral Parameter Power Series in Eigenvalue Problems

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Operator Theory, Pseudo-Differential Equations, and Mathematical Physics

Part of the book series: Operator Theory: Advances and Applications ((OT,volume 228))

Abstract

We give an overview of recent developments in Sturm-Liouville theory concerning operators of transmutation (transformation) and spectral parameter power series (SPPS). The possibility to write down the dispersion (characteristic) equations corresponding to a variety of spectral problems related to Sturm-Liouville equations in an analytic form is an attractive feature of the SPPS method. It is based on a computation of certain systems of recursive integrals. Considered as families of functions these systems are complete in the L22-space and result to be the images of the nonnegative integer powers of the independent variable under the action of a corresponding transmutation operator. This recently revealed property of the Delsarte transmutations opens the way to apply the transmutation operator even when its integral kernel is unknown and gives the possibility to obtain further interesting properties concerning the Darboux transformed Schródinger operators.

We introduce the systems of recursive integrals and the SPPS approach, explain some of its applications to spectral problems with numerical illustrations, give the definition and basic properties of transmutation operators, introduce a parametrized family of transmutation operators, study their map** properties and construct the transmutation operators for Darboux transformed Schródinger operators.

Mathematics Subject Classification (2010). Primary 34B24, 34L16, 65L15, 81Q05, 81Q60; Secondary 34L25, 34L40.

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Authors and Affiliations

  1. Department of Mathematics, CINVESTAV del IPN, Unidad Queretaro, Libramiento Norponiente No. 2000, Fracc. Real de Juriquilla, Queretaro, Qro., C.P. 76230, Mexico

    Vladislav V. Kravchenko & Sergii M. Torba

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  1. Vladislav V. Kravchenko
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  2. Sergii M. Torba
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Correspondence to Vladislav V. Kravchenko .

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Editors and Affiliations

  1. de Morelos, Facultad de Ciencias, Universidad Autónoma del Estado, Av. Universidad 1001, Col. Chamilpa, Cuernavaca, 62209, Morelos, Mexico

    Yuri I. Karlovich

  2. , Dipartimento di Matematica, Università di Torino, Via Carlo Alberto 10, Torino, 10123, Italy

    Luigi Rodino

  3. , Fakultät für Mathematik, Technische Universität Chemnitz, Reichenhainer Straße 39, Chemnitz, 09126, Germany

    Bernd Silbermann

  4. , Department of Mathematics, College of William and Mary, Jones Hall 114, Williamsburg, 23187-8795, Virginia, USA

    Ilya M. Spitkovsky

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Dedicated to 70th birthday anniversary of Prof. Dr. Vladimir S. Rabinovich.

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Kravchenko, V.V., Torba, S.M. (2013). Transmutations and Spectral Parameter Power Series in Eigenvalue Problems. In: Karlovich, Y., Rodino, L., Silbermann, B., Spitkovsky, I. (eds) Operator Theory, Pseudo-Differential Equations, and Mathematical Physics. Operator Theory: Advances and Applications, vol 228. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-0537-7_11

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