具有幺模谱的矩阵的伪谱所导致的繁荣与萧条周期
Boom and bust cycles due to pseudospectra of matrices with unimodular spectra
AI总结：
本研究揭示了特征值在单位圆上的非正规矩阵高次幂动力学的反直觉行为：无限大矩阵可指数增长，有限矩阵呈增长-衰减周期，该效应由谱与伪谱的差异决定，且存在于多种物理与数学模型中。
摘要：
我们讨论了作为恒等矩阵根的非正规矩阵的高次幂所产生的动力学，这类矩阵的所有特征值都位于单位圆上。直觉上，人们会认为随着幂次升高，这类幂次的期望值不会增长。我们证明，与直觉完全相反的行为是可能存在的。在矩阵无限大的极限下，系统可以出现指数增长。对于有限矩阵，这种指数增长是“指数增长 followed by 指数衰减”的重复周期的一部分。当矩阵的谱与其伪谱不同时，就可能出现这种效应，且指数增长率由伪谱决定。我们表明，这种效应出现在一类研究二维无相互作用系统时遇到的转移矩阵中，出现在描述Ehrenfest urn（埃伦费斯特瓮）的矩阵中，也出现在此前在阶梯型随机电路中观测到的纯度动力学中。
英文摘要：
We discuss dynamics obtained by increasing powers of non-normal matrices that are roots of the identity, and therefore have all eigenvalues on the unit circle. Naively, one would expect that the expectation value of such powers cannot grow as one increases the power. We demonstrate that, rather counterintuitively, a completely opposite behavior is possible. In the limit of infinitely large matrices one can have an exponential growth. For finite matrices this exponential growth is a part of repeating cycles of exponential growths followed by exponential decays. The effect can occur if the spectrum is different than the pseudospectrum, with the exponential growth rate being given by the pseudospectrum. We show that this effect appears in a class of transfer matrices appearing in studies of two-dimensional non-interacting systems, for a matrix describing the Ehrenfest urn, as well as in previously observed purity dynamics in a staircase random circuit.
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