The Time Fractional Schrödinger Equation on Hilbert Space

Abstract

Abstract. We study the linear fractional Schrödinger equation on a Hilbert space, with a fractional time derivative of order 0 < α < 1, and a self-adjoint generator A. Using the spectral theorem we prove existence and uniqueness of strong solutions, and we show that the solutions are governed by an operator solution family {Uα(t)} t≥0 . Moreover, we prove that the solution family Uα(t) converges strongly to the family of unitary operators e −itA , as α approaches to 1.Mathematics Subject Classification. 35Q41, 35R11, 81S99; Secondary 47D08, 47D03, 81C10.

The Probabilistic Point of View on the Generalized Fractional Partial Differential Equations

“…Fractional Schrödinger equation is getting popularity in physics community, see e.g. [48], [61], [10], [17] and references therein. For the fractional versions of the wave equations we refer to [66].…”

On Lie Symmetry Analysis of Certain Coupled Fractional Ordinary Differential Equations

“…However, the derivation of the exact solution of FDEs is not an easy task since the properties of a fractional derivative are harder than the classical derivative. Recently, several research groups have been developed to derive the exact and numerical solutions of FDEs such as invariant subspace method [11,21,39,[51][52][53]57,58,71], variational iteration method [44], homotopy perturbation method [45], operational matrix method [55], collocation method [40] and so on [14,23,68,69]. Among those methods, the Lie symmetry analysis method is an algorithmic approach that provides an efficient tool to construct an exact solution of FDEs in a systematic way.…”

Regularity of the solution for a final value problem for the Rayleigh‐Stokes equation

“…These applications appears in gravitation elastic membrane, electrostatics, fluid flow, steady state, heat conduction and many other topics. In the recent years, there has been a significant development in partial differential equations involving fractional derivatives, see for instance, time fractional diffusion equations as in other works, [1][2][3] fractional Navier-Stokes equations as in Zhou and Peng,4,5 fractional Schroinger equations as in other works, [6][7][8] diffusion equation, and cable equation. [9][10][11][12][13] In this paper, we consider the Rayleigh-Stokes problem for a generalized second-grade fluid with a fractional derivative model as follows:…”