The time fractional Schr\"odinger equation on Hilbert space

TL;DR

This paper investigates the fractional Schr"odinger equation on Hilbert space with a fractional time derivative, establishing existence, uniqueness, and convergence of solutions to the classical case as the fractional order approaches one.

Contribution

It introduces a spectral theorem-based framework for solving the fractional Schr"odinger equation and demonstrates the strong convergence of solutions to the classical unitary evolution.

Findings

Existence and uniqueness of strong solutions for the fractional Schr"odinger equation.

Construction of an operator solution family _{lpha}(t) for the equation.

Strong convergence of _{lpha}(t) to e^{-itA} as pproaches 1.

Abstract

We study the linear fractional Schr\"odinger equation on a Hilbert space, with a fractional time derivative of order $0<\alpha<1,$ 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 $\{U_{\alpha}(t)\}_{t\geq 0}$. Moreover, we prove that the solution family $U_{\alpha}(t)$ converges strongly to the family of unitary operators $e^{-itA},$ as $\alpha$ approaches to $1$.