有限间隙雅可比矩阵，III。超越SzegIgnClass
Finite Gap Jacobi Matrices, III. Beyond the Szegő Class
摘要：
设$\fre\subset\bbR$是$\ell+1$不相交闭区间的有限并，用$ω_j$表示$j$最左带的调和测度。$\fre$的频率模块是$ω_1,..., ω_\ell$的所有整数组合的集合。设$\{\tilde{a}_n, \tilde{b}_n\}_{n=1}^\infty$是$\fre$的等谱环面上的一点，$\tilde{p}_n$是其正交多项式。设$\{a_n,b_n\}_{n=1}^\infty$是具有$a_n = \tilde{a}_n + δa_n$、$b_n = \tilde{b}_n + δb_n$的半线雅可比矩阵。假设\[ \sum_{n=1}^\infty %(\abs{a_n-\tilde{a}_n}^2 + \abs{b_n-\tilde{b}_n}^2) <\infty \abs{δa_n}^2 + \abs{δb_n}^2 <\infty \]和$\sum_{n=1}^N e^{2πiωn} δa_n$，对于频率模块中的所有$ω$，$\sum_{n=1}^N e^{2πiωn} δb_n$具有与$N\to\infty$相同的有限极限。此外，如果这些部分和关于$ω$至多亚指数地增长，则对于$z\in\bbC\setminus\bbR$，$p_n(z)/\tilde{p}_n(z)$具有与$n\to\infty$相同的极限。此外，我们表明，有非SzegIgnclass $J$的，这是成立的。
英文摘要：
Let $\fre\subset\bbR$ be a finite union of $\ell+1$ disjoint closed intervals and denote by $ω_j$ the harmonic measure of the $j$ leftmost bands. The frequency module for $\fre$ is the set of all integral combinations of $ω_1,..., ω_\ell$. Let $\{\tilde{a}_n, \tilde{b}_n\}_{n=1}^\infty$ be a point in the isospectral torus for $\fre$ and $\tilde{p}_n$ its orthogonal polynomials. Let $\{a_n,b_n\}_{n=1}^\infty$ be a half-line Jacobi matrix with $a_n = \tilde{a}_n + δa_n$, $b_n = \tilde{b}_n + δb_n$. Suppose \[ \sum_{n=1}^\infty %(\abs{a_n-\tilde{a}_n}^2 + \abs{b_n-\tilde{b}_n}^2) <\infty \abs{δa_n}^2 + \abs{δb_n}^2 <\infty \] and $\sum_{n=1}^N e^{2πiωn} δa_n$, $\sum_{n=1}^N e^{2πiωn} δb_n$ have finite limits as $N\to\infty$ for all $ω$ in the frequency module. If, in addition, these partial sums grow at most subexponentially with respect to $ω$, then for $z\in\bbC\setminus\bbR$, $p_n(z)/\tilde{p}_n(z)$ has a limit as $n\to\infty$. Moreover, we show that there are non-Szegő class $J$'s for which this holds.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
