Access the full text.
Sign up today, get DeepDyve free for 14 days.
Loading next page...
/lp/springer-journals/schr-dinger-equations-with-fractional-laplacians-6kAERvZlRx
References (4)
-
R. Blumenthal, R. Getoor, D. Ray (1961)
On the distribution of first hits for the symmetric stable processes.Transactions of the American Mathematical Society, 99
-
G. Kallianpur, D. Kannan, R. Karandikar (1985)
Analytic and sequential Feynman integrals on abstract Wiener and Hilbert spaces, and a Cameron-Martin formulaAnnales De L Institut Henri Poincare-probabilites Et Statistiques, 21
-
I. Podlubny (1999)
Fractional differential equations : an introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications -
R. Getoor (1961)
First passage times for symmetric stable processes in spaceTransactions of the American Mathematical Society, 101
- Publisher
- Springer Journals
- Copyright
- Copyright © Springer-Verlag New York Inc. 2000
- Subject
- Mathematics; Calculus of Variations and Optimal Control; Optimization; Systems Theory, Control; Theoretical, Mathematical and Computational Physics; Mathematical Methods in Physics; Numerical and Computational Physics, Simulation
- ISSN
- 0095-4616
- eISSN
- 1432-0606
- DOI
- 10.1007/s002450010014
- Publisher site
- See Article on Publisher Site
Abstract
. It is shown that the unique solution of\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\begin{cases}{\frac{\partial }{\partial t}}\psi & (t, x) = -(z^2)^{\alpha/2}(-triangle)^{\alpha/2}\psi(t, x)+V(z, x)\psi(t, x),\\\psi& (0,x) = f(x), \end{cases}$$\end{document}} can be represented as {\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\Psi(t,x)=\Bbb E f(x+(z)^{1/\alpha}X_s)\exp\left\{ \int_0^t V(z, x+(z)^{1\alpha} X_u)\,du\right\},$$\end{document}} where X=(Xt , t≥ 0)is a stable process whose generator is (-Δ)α/2with X0=0 .
Journal
Applied Mathematics & Optimization – Springer Journals
Published: Nov 1, 2000
Keywords: Fractional Laplacian; Schrödinger equations; Analytic continuation; Symmetric stable processes; Feynman—Kac formula; Exponential integrability; Markov property; AMS Classification. Primary 60H05, 60H10, Secondary 90A09, 90A12.