Time-dependent Schrödinger perturbations of transition densities

Krzysztof Bogdan¹ ; Wolfhard Hansen² ; Tomasz Jakubowski¹

¹ Institute of Mathematics and Computer Science Wrocław University of Technology Wybrzeże Wyspiańskiego 27 50-370 Wrocław, Poland
² Fakultät für Mathematik Universität Bielefeld Postfach 100131 D-33501 Bielefeld, Germany

Studia Mathematica, Tome 189 (2008) no. 3, pp. 235-254

Résumé

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.

Détail

DOI : 10.4064/sm189-3-3

Keywords: construct fundamental solution partial t mit delta y functions certain integral space time relative smallness particular those satisfying relative kato condition resulting transition density comparable gaussian kernel finite time even asymptotically equal gaussian kernel small time under relative kato condition result generalized arbitrary strictly positive finite time nonhomogeneous transition densities measure spaces discuss specific applications schr dinger perturbations fractional laplacian view the theorem holds fundamental solution corresponding operator