标题： 一维薛定谔方程的条件对称性和谱 显示英文标题

标题： Conditional symmetry and spectrum of the one-dimensional Schrödinger equation

摘要： 我们开发了一种代数方法，用于研究一维定态薛定谔方程的谱性质，该方法基于其高阶条件对称性。 这种方法可以显式地用$n\times n$矩阵表示薛定谔算子，对于任何$n\in {\bf N}$，从而将谱问题转化为纯粹的代数问题，即寻找常数$n\times n$矩阵的特征值。 讨论了与所谓的准精确可解模型的联系。 特别是确定，当条件对称性退化为高阶李对称性时，对应于精确可解的薛定谔方程。 对允许非平凡高阶李对称性的薛定谔方程进行了对称分类，这产生了一个精确可解薛定谔方程的层次结构。 这些方程的精确解以显式形式构造出来。 简要讨论了所开发技术在多维线性和一维非线性薛定谔方程中的可能应用。 显示英文摘要

摘要： We develop an algebraic approach to studying the spectral properties of the stationary Schr\"odinger equation in one dimension based on its high order conditional symmetries. This approach makes it possible to obtain in explicit form representations of the Schr\"odinger operator by $n\times n$ matrices for any $n\in {\bf N}$ and, thus, to reduce a spectral problem to a purely algebraic one of finding eigenvalues of constant $n\times n$ matrices. The connection to so called quasi exactly solvable models is discussed. It is established, in particular, that the case, when conditional symmetries reduce to high order Lie symmetries, corresponds to exactly solvable Schr\"odinger equations. A symmetry classification of Sch\"odinger equation admitting non-trivial high order Lie symmetries is carried out, which yields a hierarchy of exactly solvable Schr\"odinger equations. Exact solutions of these are constructed in explicit form. Possible applications of the technique developed to multi-dimensional linear and one-dimensional nonlinear Schr\"odinger equations is briefly discussed.