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 Schrodinger 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 Pade--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 Schrodinger equation into the Helmholtz equation with a suitable density.

中文翻译：

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薛定谔方程的谱求和规则

我们研究 $Z(s) = \sum_n E_n^{-s}$ 形式的求和规则，其中 $E_n$ 是与时间无关的薛定谔方程（在一维或多维中）和 $s$ 的特征值是级数收敛的有理数。我们已经使用微扰理论获得了微扰中高达二阶的求和规则的明确公式，并且我们已经通过 Pade-近似非微扰地扩展了它。对于在一维上用一种杂质装饰的盒子的特殊情况，我们精确地计算了整数阶的前几个求和规则；一阶求和法则也精确地计算了一个有两种杂质的盒子的问题。在二维中，我们考虑了杂质分布在任意半径圆上的情况，并计算了二阶精确求和规则。