Dispersive estimates for time and space fractional Schrödinger equations. (English) Zbl 1475.35398
Summary: In this paper, we consider the Cauchy problem for the fractional Schrödinger equation \(iD^\alpha_t u - (- (\Delta)^\frac{\beta}{2} u = 0 \) with
\(0 < \alpha < 1\), \(\beta > 0\). We establish the dispersive estimates by a carefully study of the Mittag-Leffler functions and give some applications as well. In particular, we prove that the decay rates are sharp.
MSC:
| 35R11 | Fractional partial differential equations |
| 26A33 | Fractional derivatives and integrals |
| 35B40 | Asymptotic behavior of solutions to PDEs |
| 35B45 | A priori estimates in context of PDEs |
| 35Q41 | Time-dependent Schrödinger equations and Dirac equations |
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