Time-dependent Schrödinger perturbations of transition densities

Tom 189 / 2008

Krzysztof Bogdan, Wolfhard Hansen, Tomasz Jakubowski Studia Mathematica 189 (2008), 235-254 MSC: 47A55, 60J35, 60J57. DOI: 10.4064/sm189-3-3

Streszczenie

We construct the fundamental solution of $\partial _t-{\mit\Delta} _y- q(t,y)$ for functions $q$ with a certain integral space-time relative smallness, in particular for those satisfying a relative Kato condition. The resulting transition density is comparable to the Gaussian kernel in finite time, and it is even asymptotically equal to the Gaussian kernel (in small time) under the relative Kato condition.

The result is generalized to arbitrary strictly positive and finite time-nonhomogeneous transition densities on measure spaces.

We also discuss specific applications to Schrödinger perturbations of the fractional Laplacian in view of the fact that the 3P Theorem holds for the fundamental solution corresponding to the operator.

Autorzy

  • Krzysztof BogdanInstitute of Mathematics and Computer Science
    Wrocław University of Technology
    Wybrzeże Wyspiańskiego 27
    50-370 Wrocław, Poland
    e-mail
  • Wolfhard HansenFakultät für Mathematik
    Universität Bielefeld
    Postfach 100131
    D-33501 Bielefeld, Germany
    e-mail
  • Tomasz JakubowskiInstitute of Mathematics and Computer Science
    Wrocław University of Technology
    Wybrzeże Wyspiańskiego 27
    50-370 Wrocław, Poland
    e-mail

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