分形Laplacian谱的外逼近
Outer Approximation of the Spectrum of a Fractal Laplacian
摘要：
我们提出了一种新的方法来近似的Neumann谱的拉普拉斯算子上的分形K在平面上作为一个重整化的限制的Neumann谱的标准拉普拉斯算子上的一个序列的域，近似K从外面。该方法允许一个数值逼近的本征值和本征函数的较低部分的频谱。我们目前的实验证据表明，该方法的工作原理，通过查看的分形拉普拉斯算子的频谱是已知的（单位间隔和谢尔宾斯基垫片（SG））的例子。我们还提出了一个投机性的描述的频谱上的标准谢尔宾斯基地毯（SC），存在一个自相似的拉普拉斯算子是已知的，也对非对称和随机地毯和八角形，存在一个自相似的拉普拉斯算子是未知的。目前，我们还没有解释为什么这种方法应该工作。尽管如此，我们能够证明一些新的结果的光谱结构涉及“小型化”的本征函数，我们发现通过检查使用我们的方法获得的实验结果。
英文摘要：
We present a new method to approximate the Neumann spectrum of a Laplacian on a fractal K in the plane as a renormalized limit of the Neumann spectra of the standard Laplacian on a sequence of domains that approximate K from the outside. The method allows a numerical approximation of eigenvalues and eigenfunctions for lower portions of the spectrum. We present experimental evidence that the method works by looking at examples where the spectrum of the fractal Laplacian is known (the unit interval and the Sierpinski Gasket (SG)). We also present a speculative description of the spectrum on the standard Sierpinski carpet (SC), where existence of a self-similar Laplacian is known, and also on nonsymmetric and random carpets and the octagasket, where existence of a self-similar Laplacian is not known. At present we have no explanation as to why the method should work. Nevertheless, we are able to prove some new results about the structure of the spectrum involving "miniaturization" of eigenfunctions that we discovered by examining the experimental results obtained using our method.
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