精确可解非相对论性问题类的扩充
Enlarging the class of exactly solvable nonrelativistic problems
摘要：
我们解除了对角表示的哈密顿量的约束，通过搜索平方可积的基础，支持三对角矩阵表示的波算子。这样做的结果是精确可解的问题，一类潜在的是大于，和/或推广，什么是已知的。此外，我们还发现了一些已知势的解空间的新表示。这个问题转化为寻找波函数展开系数的三项递归关系的解。其中一些解是一类新的正交多项式。给出的例子，这不是详尽无遗的，是在一维和三维的问题。用这种方法得到的解析解既包括哈密顿量的离散谱，也包括连续谱。
英文摘要：
We lift the constraint of a diagonal representation of the Hamiltonian by searching for square integrable bases that support a tridiagonal matrix representation of the wave operator. Doing so results in exactly solvable problems with a class of potentials which is larger than, and/or generalization of, what is already known. In addition, we found new representations for the solution space of some well known potentials. The problem translates into finding solutions of the resulting three-term recursion relation for the expansion coefficients of the wavefunction. Some of these solutions are new kind of orthogonal polynomials. The examples given, which are not exhaustive, are for problems in one and three dimensions. The analytic solutions obtained by this approach include the discrete as well as the continuous spectrum of the Hamiltonian.
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