论文标题
标量和矩阵SchrödingerCocycles的非裂解行为集
Sets of non-Lyapunov behaviour for scalar and matrix Schrödinger cocycles
论文作者
论文摘要
我们在一个维度上讨论了与Ergodic离散的Schrödinger运算符相关的符号传递矩阵的奇异值的增长，并具有标量和基质值势。尽管对于频谱参数的个体值，但指数增长的速率几乎肯定由Lyapunov指数控制，但对于参数的所有值，这通常不是同时的。特殊集合的结构本身就是有趣的，并且在操作员的光谱分析中也很重要。我们提出了新的结果以及几个较旧的结果的扩增和概括，还列出了一些开放问题。 这是两个样本结果。在负面的一面，对于任何正方形的序列$ p_n $，在频谱中都有一组残留的能量集，中间值（$ w $ the $ 2W $中的$ w $ th）的增长速度不超过$ p_n^{ - 1} $。在正面，对于包括I.I.D的大量共生那些，奇数值的增长的一组能量不是由lyapunov指数给出的，相对于任何规格函数$ρ（t）$，零hausdorff的量度为零。 次谐函数理论中采用的论点也产生了无独立的利益的概括：对于每个$ k $，第一个$ k $ lyapunov指数的平均值是概率措施的对数潜力。
We discuss the growth of the singular values of symplectic transfer matrices associated with ergodic discrete Schrödinger operators in one dimension, with scalar and matrix-valued potentials. While for an individual value of the spectral parameter the rate of exponential growth is almost surely governed by the Lyapunov exponents, this is not, in general, true simultaneously for all the values of the parameter. The structure of the exceptional sets is interesting in its own right, and is also of importance in the spectral analysis of the operators. We present new results along with amplifications and generalisations of several older ones, and also list a few open questions. Here are two sample results. On the negative side, for any square-summable sequence $p_n$ there is a residual set of energies in the spectrum on which the middle singular value (the $W$-th out of $2W$) grows no faster than $p_n^{-1}$. On the positive side, for a large class of cocycles including the i.i.d. ones, the set of energies at which the growth of the singular values is not as given by the Lyapunov exponents has zero Hausdorff measure with respect to any gauge function $ρ(t)$ such that $ρ(t)/t$ is integrable at zero. The employed arguments from the theory of subharmonic functions also yield a generalisation of the Thouless formula, possibly of independent interest: for each $k$, the average of the first $k$ Lyapunov exponents is the logarithmic potential of a probability measure.
