周期图上薛定谔算子的渐近等谱性

IF 1.4 3区 数学 Q1 MATHEMATICS
Natalia Saburova
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引用次数: 0

摘要

我们考虑周期图上具有周期势的离散薛定谔算子。它们的谱由有限数量的带组成。我们通过周期性地添加边(不改变顶点集)来扰动周期图,并证明如果添加的边足够长,那么扰动后的图与某个维数更高但没有长边的周期图渐近等谱。我们还得到了一个标准,即扰动图不仅渐近等谱,而且与这个维度更高的周期图完全等谱。这种渐近等谱周期图的最简单例子之一是长边扰动的方格和立方格。当添加的边的长度趋于无穷大时,我们还可以得到扰动图上薛定谔算子谱带端点的渐近线。
本文章由计算机程序翻译,如有差异,请以英文原文为准。

Asymptotic isospectrality of Schrödinger operators on periodic graphs

Asymptotic isospectrality of Schrödinger operators on periodic graphs

Asymptotic isospectrality of Schrödinger operators on periodic graphs

We consider discrete Schrödinger operators with periodic potentials on periodic graphs. Their spectra consist of a finite number of bands. We perturb a periodic graph by adding edges in a periodic way (without changing the vertex set) and show that if the added edges are long enough, then the perturbed graph is asymptotically isospectral to some periodic graph of a higher dimension but without long edges. We also obtain a criterion for the perturbed graph to be not only asymptotically isospectral but just isospectral to this higher dimensional periodic graph. One of the simplest examples of such asymptotically isospectral periodic graphs is the square lattice perturbed by long edges and the cubic lattice. We also get asymptotics of the endpoints of the spectral bands for the Schrödinger operator on the perturbed graph as the length of the added edges tends to infinity.

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来源期刊
Analysis and Mathematical Physics
Analysis and Mathematical Physics MATHEMATICS, APPLIED-MATHEMATICS
CiteScore
2.70
自引率
0.00%
发文量
122
期刊介绍: Analysis and Mathematical Physics (AMP) publishes current research results as well as selected high-quality survey articles in real, complex, harmonic; and geometric analysis originating and or having applications in mathematical physics. The journal promotes dialog among specialists in these areas.
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