薛定谔方程可积性的伽罗瓦方法
Galoisian Approach to integrability of Schrödinger Equation
摘要：
本文从微分伽罗瓦理论的角度研究了非相对论定态薛定谔方程。主要的算法工具是二阶常线性微分算子的拉回，以实现有理函数系数（“代数化”），以及求解所得方程的Kovacic算法。特别是，我们使用这种伽罗瓦方法来分析达布变换，Crum迭代和超对称量子力学。我们得到了一大类薛定谔方程的基态、本征值、本征函数、本征态和微分伽罗瓦群，例如，具有精确可解和形状不变势的方程（这些项在文中定义）。最后，我们介绍了一种方法来确定何时精确可解性是可能的。
英文摘要：
In this paper, we examine the non-relativistic stationary Schrödinger equation from a differential Galois-theoretic perspective. The main algorithmic tools are pullbacks of second order ordinary linear differential operators, so as to achieve rational function coefficients ("algebrization"), and Kovacic's algorithm for solving the resulting equations. In particular, we use this Galoisian approach to analyze Darboux transformations, Crum iterations and supersymmetric quantum mechanics. We obtain the ground states, eigenvalues, eigenfunctions, eigenstates and differential Galois groups of a large class of Schrödinger equations, e.g. those with exactly solvable and shape invariant potentials (the terms are defined within). Finally, we introduce a method for determining when exact solvability is possible.
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