Dirac方程和薛定谔方程的束缚态计算
On computing bound states of the Dirac and Schrödinger Equations
摘要：
我们将计算束缚态的量子化学问题转化为求解一族参数化紧致积分算子的辅助本征值问题。算子的紧性保证了它们的谱是离散的，并且有界的，唯一可能的聚点为零。我们表明，通过改变参数，我们总是可以找到束缚态，即，满足原始方程且可归一化的本征函数。虽然对于非相对论性方程，这些性质可能并不令人惊讶，但值得注意的是，这同样适用于原始相对论性算子的谱没有下界的相对论性方程。我们证明，从任意初始化的迭代导致的解决方案，所规定的紧算子的属性。
英文摘要：
We cast the quantum chemistry problem of computing bound states as that of solving a set of auxiliary eigenvalue problems for a family of parameterized compact integral operators. The compactness of operators assures that their spectrum is discrete and bounded with the only possible accumulation point at zero. We show that, by changing the parameter, we can always find the bound states, i.e., the eigenfunctions that satisfy the original equations and are normalizable. While for the non-relativistic equations these properties may not be surprising, it is remarkable that the same holds for the relativistic equations where the spectrum of the original relativistic operators does not have a lower bound. We demonstrate that starting from an arbitrary initialization of the iteration leads to the solution, as dictated by the properties of compact operators.
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