The Time Fractional Schrödinger Equation on Hilbert Space

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• 作者：Przemysław Górka ; Humberto Prado ; Juan Trujillo
• 关键词：Fractional quantum mechanics ; Caputo derivative ; Schrödinger equation
• 刊名：Integral Equations and Operator Theory
• 出版年：2017
• 出版时间：January 2017
• 年：2017
• 卷：87
• 期：1
• 页码：1-14
• 全文大小：522KB
• 刊物类别：Mathematics and Statistics
• 刊物主题：Analysis;
• 出版者：Springer International Publishing
• ISSN：1420-8989
• 卷排序：87

文摘

We study the linear fractional Schrödinger 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 \(\{ U_{\alpha }(t)\}_{t\ge 0}\). 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.