數學物理方程之可解性

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Title:	數學物理方程之可解性
Other Titles:	Solvability of Equations in Mathematical Physics
Authors:	何俊麟
Contributors:	淡江大學物理學系
Date:	2010
Issue Date:	2011-07-05 23:38:43 (UTC+8)
Abstract:	我們將研究下列問題: 1. 我們將嘗試將初始位勢法推廣到其他的重要的數學物理方程, 希望此法也能對這些方程的精確及準精確可解性提供一個較簡潔的處理方案。我們準備考慮的方程包括: 相對性的狄拉克(Dirac) 方程, 描述隨機現象的福克-普朗克(Fokker-Planck) 方程, 及多粒子系統的薛丁格(Schrodinger) 方程。 2. 我們將繼續探討與新的X特異正交多項式相關的研究課題。我們將探討它們的物理應用, 建構與之相關的物理模型, 及它們可能的進一步的數學推廣。 3. 我們將繼續探討量子系統在周期外場重的穩定性問題。我們將考慮其他能以解析法, 近似法, 或數值法處理的位壘, 以及不同的場振盪模式。 4. 我們將更深入的探討奇性位勢中可包容束敷態的奇特例子。 5. 我們將進一步研究一類由我們導出的離散FokkerPlanck 方程的性質。 We intend to investigate the following problems: 1. we shall generalize the prepotential approach to other important equations in mathematical physics. The equations we have in mind include: Dirac equation, Fokker-Planck equation, and multi-particle Schrodinger equation; 2. we shall investigate topics related to the newly discovered X exceptional orthogonal polynomials. These include their possible physical applications, and further mathematical generalizations; 3. we will also continue our investigations of the stability problem of a quantum system in time-dependent fields which could be treated either analytically, approximately, or numerically; 4. we shall study more deeply the strange cases of bound states in singular potentials; 5. we shall also study the properties of the new class of discrete Fokker-Planck equations derived by us recently.
Appears in Collections:	[Graduate Institute & Department of Physics] Research Paper

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