非均匀鼓的精确求和规则
Exact sum rules for inhomogeneous drums
摘要：
本文导出了任意形状、任意密度的圆筒在不同边界条件下本征值求和规则的一般表达式。我们提出的公式是一个推广的一维非齐次系统的类似公式，我们已经得到了在以前的文件。我们还讨论了这些公式的扩展到更高的维度。我们表明，在密度只取决于一个变量的特殊情况下，任何整数阶的求和规则可以表示在一个单一的系列。作为我们的结果的应用程序，我们得到了精确的和规则的均匀的圆环与不同的边界条件，均匀的圆形部门和径向非均匀的圆环与Dirichlet边界条件。
英文摘要：
We derive general expressions for the sum rules of the eigenvalues of drums of arbitrary shape and arbitrary density, obeying different boundary conditions. The formulas that we present are a generalization of the analogous formulas for one dimensional inhomogeneous systems that we have obtained in a previous paper. We also discuss the extension of these formulas to higher dimensions. We show that in the special case of a density depending only on one variable the sum rules of any integer order can be expressed in terms of a single series. As an application of our result we derive exact sum rules for the a homogeneous circular annulus with different boundary conditions, for a homogeneous circular sector and for a radially inhomogeneous circular annulus with Dirichlet boundary conditions.
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