AI 中文总结
研究双边托达晶格在精确谱相空间的整体适定性，通过定义相空间\(\mathcal Q\)，证明\(q\in\mathcal Q\)时托达晶格有经典解，且解具有唯一性、保持在\(\mathcal Q\)中并连续依赖初始数据。
AI 中文摘要
我们确定了双边托达晶格的精确谱相空间。设\(q = \{a_n, b_n\}_{n\in\mathbb Z}\)为左右半直线雅可比算子的系数，其谱测度记为\(\sigma_{\pm}^{q}\)。定义相空间\(\mathcal Q=\left\{ \begin{array} [c]{c}% q=\{a_n,b_n\}_{n\in\mathbb Z}: a_n>0,\ b_{n} \in \mathbb{R} \text{ 且对每个 }c>0 有 \int_{\mathbb R}e^{c|\lambda|}\sigma^q_\pm(d\lambda)<\infty \end{array} \right\}\)。可积性条件使表示测度唯一。我们证明\(q\in\mathcal Q\)当且仅当具有初始数据\(q\)的托达晶格对所有正负时间都有经典解。此外，解保持在\(\mathcal Q\)中，是唯一的，且在紧致时间区间上关于初始数据连续依赖。
英文摘要
We identify an exact spectral phase space for the two-sided Toda lattice. Let $q=\{a_n,b_n\}_{n\in\mathbb Z}$ be coefficients of the right and left half-line Jacobi operators and denote their spectral measures by $σ_{\pm}^{q}$. Define a phase space \[ \mathcal Q=\left\{ \begin{array} [c]{c}% q=\{a_n,b_n\}_{n\in\mathbb Z}: a_n>0,\ b_{n} \in \mathbb{R} \text{ and} \int_{\mathbb R}e^{c|λ|}σ^q_\pm(dλ)<\infty \text{ for every }c>0 \end{array} \right\} . \] The integrability condition makes the representing measures unique. We prove that $q\in\mathcal Q$ if and only if the Toda lattice with initial datum $q$ admits a classical solution for all positive and negative times. Moreover, the solution remains in $\mathcal Q$, is unique, and depends continuously on the initial datum, uniformly on compact time intervals.
