Mathematical Sciences: Harmonic Analysis, Special Functions and Separation of Variables

分离变量是最好的已知方法， 偏微分方程的简化形式 它是在 本科数学课程，往往被视为没有什么 不仅仅是意外 事实上， 分离变量的数学和物理原因 以及为什么它实际上适用于一大类 方程 这项工作的目的是继续长期 调查情况和适用方法 处理可以分解的方程。 这项工作分为几个部分。 首先是问题 确定变量分离的一般定义， 适用于所有偏微分方程（系统）， 提出可能分离类型的结构理论。 第二个问题是可分性的内在刻画问题 坐标 对于Hamilton-Jacobi和Schrodinger方程， 问题基本解决了。 对于方程组，例如 狄拉克方程和线性弹性方程，它仍然是开放的。 工作 将继续在这一领域。 最后，还有一个问题， 确定当变量为 在一个特定的方程中分离，并建立 这些功能。 特别强调的是 属性是一个后果的对称性 关联方程

Separation of variables is the best known method of reducing partial differential equations to simpler form. It is taught in undergraduate mathematics courses and is often viewed as nothing more than a serendipitous accident. There are, in fact, deep mathematical and physical reasons why separation of variables works and why it actually applies across a large class of equations. The purpose of this work is to continue a long-term investigation into the circumstances and applicable methods for treating equations which can be decomposed. The work divides into several parts. First is the problem of determining a general definition of variable separation that applies to all (systems of) partial differential equations and working out the structure theory of possible separation types. Second is the problem of intrinsic characterization of separable coordinates. For Hamilton-Jacobi and Schrodinger equations this problem is essentially solved. For systems of equations such as the Dirac and linear elasticity equations, it remains open. Work will continue in this area. Finally, there is the problem of determining the special functions that arise when variables are separated in a particular equation and establishing properties of these functions. Particular emphasis is placed on those properties which are a consequence of the symmetry of the associated equation.