论文标题
Schrödinger方程的光谱总和规则
Spectral sum rules for the Schrödinger equation
论文作者
论文摘要
我们研究$ z（s）= \ sum_n e_n^{ - s} $的总和规则，其中$ e_n $是当时的特征值 - 独立的schrödinger方程（一个或多个维度），而$ s $是串联收益的合理数量。我们已经使用扰动理论获得了扰动中总和最高为二阶的显式公式，并且我们通过帕德（Approximant）进行了非扰动。对于一个在一个维度上装饰有一个杂质的盒子的特殊情况，我们已经准确地计算了整数的前几个总和规则；订单的总规则也已完全计算出具有两个杂质的盒子问题。在两个维度中，我们考虑了分布在任意半径圆圈上的杂质的情况，并且我们计算了第二顺序的确切总和规则。最后，我们证明可以通过将schrödinger方程转换为具有合适密度的helmholtz方程来获得确切的总和规则。
We study the sum rules of the form $Z(s) = \sum_n E_n^{-s}$, where $E_n$ are the eigenvalues of the time--independent Schrödinger equation (in one or more dimensions) and $s$ is a rational number for which the series converges. We have used perturbation theory to obtain an explicit formula for the sum rules up to second order in the perturbation and we have extended it non--perturbatively by means of a Padé--approximant. For the special case of a box decorated with one impurity in one dimension we have calculated the first few sum rules of integer order exactly; the sum rule of order one has also been calculated exactly for the problem of a box with two impurities. In two dimensions we have considered the case of an impurity distributed on a circle of arbitrary radius and we have calculated the exact sum rules of order two. Finally we show that exact sum rules can be obtained, in one dimension, by transforming the Schrödinger equation into the Helmholtz equation with a suitable density.
