COMPUTING SPECTRA WITHOUT SOLVING EIGENVALUE PROBLEMS
COMPUTING SPECTRA WITHOUT SOLVING EIGENVALUE PROBLEMS
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DOI：
10.1137/17m1156721
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发表时间：
2019-01-01
影响因子：
3.1
通讯作者：
Mayboroda, Svitlana
中科院分区：
文献类型：
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作者：
Arnold, Douglas N.;David, Guy;Mayboroda, Svitlana
椭圆算子的本征值和本征函数的近似是应用数学和计算物理中的一个重要计算任务。一个重要的例子，特别是在量子物理学中，是计算具有无序势的薛定谔算符的谱。与平面波或布洛赫波不同的是，对于周期势和其他有序势，本征函数是以薛定谔本征函数的形式出现的，对于许多形式的无序势，本征函数基本上保持在初始域的一个非常小的子集中。一个著名的例子是安德森本地化，其中，在一个连续的版本中，潜在的是一个分段常数函数的均匀网格，其值是独立于均匀随机分布采样。我们在这里提出了一个新的方法来近似的特征值和支持这种本地化的本征函数的子区域。这种方法是基于最近的理论工具的本地化景观和有效的潜力。该方法是确定性的，在这个意义上，近似值是基于对随机势的特定实现的检查来计算的，并且预测敏感地依赖于特定实现的量，而不是提供关于与具有特定分布的势族相关联的谱的统计或概率结果。这些方法，这只在理论上得到了部分证明，使计算的位置和形状的近似支持的本征函数，近似值的许多本征值，本征值计数函数和密度的状态，所有的成本解决一个单一的源问题，为同一椭圆算子。我们研究的有效性和局限性的方法，通过广泛的计算在一个和两个维度，使用各种分段恒定的潜力与采样值从各种不同的相关或不相关的随机分布。
The approximation of the eigenvalues and eigenfunctions of an elliptic operator is a key computational task in many areas of applied mathematics and computational physics. An important case, especially in quantum physics, is the computation of the spectrum of a Schrodinger operator with a disordered potential. Unlike plane waves or Bloch waves that arise as Schrodinger eigenfunctions for periodic and other ordered potentials, for many forms of disordered potentials the eigenfunctions remain essentially localized in a very small subset of the initial domain. A celebrated example is Anderson localization, for which, in a continuous version, the potential is a piecewise constant function on a uniform grid whose values are sampled independently from a uniform random distribution. We present here a new method for approximating the eigenvalues and the subregions which support such localized eigenfunctions. This approach is based on the recent theoretical tools of the localization landscape and effective potential. The approach is deterministic in the sense that the approximations are calculated based on the examination of a particular realization of a random potential, and predict quantities that depend sensitively on the particular realization, rather than furnishing statistical or probabilistic results about the spectrum associated to a family of potentials with a certain distribution. These methods, which have only been partially justified theoretically, enable the calculation of the locations and shapes of the approximate supports of the eigenfunctions, the approximate values of many of the eigenvalues, and of the eigenvalue counting function and density of states, all at the cost of solving a single source problem for the same elliptic operator. We study the effectiveness and limitations of the approach through extensive computations in one and two dimensions, using a variety of piecewise constant potentials with values sampled from various different correlated or uncorrelated random distributions.