AI 中文总结
研究将标量逻辑斯谛迭代表示为乘法算子，通过解析控制和数值细化研究其在特定参数下的固定矩阵元，包括不同\(r\)值的情况及相关诊断，还研究了有限维递归，为量子逻辑斯谛映射研究提供新方法和结果。
AI 中文摘要
我们将标量逻辑斯谛迭代表示为\(L^2([0,1])\)上的乘法算子，并研究其在归一化移位勒让德基下的固定矩阵元。三种参数范围允许进行解析控制。在\(r = 5/2\)时，每个固定矩阵元收敛到\(3\delta_{kl}/5\)；在\(r = 16/5\)时吸引的周期二轨道给出了偶数和奇数子序列的相位分辨极限；在\(r = 4\)时，精确的切比雪夫矩表示给出了迭代\(n\)时每个固定矩阵元到\(\delta_{kl}/2\)的\(O(4^{-n})\)逼近。\(r = 37/10\)的情况仅通过受控有限数值细化研究处理。还包括互补的有限诊断，如矩阵元时间依赖性、分岔式图等。我们还研究了具有固定矩阵\(R\)的单独有限维递归\(X_{k + 1}=R X_k(I - X_k)R^\dagger\)。这些算子值计算是探索性有限时间数值。解析陈述涉及指定基指标的固定矩阵元，与有限分辨率观测不同，并不意味着算子范数收敛。还包括正则化相空间提升作为受控可视化。
英文摘要
We represent scalar logistic iterates as multiplication operators on $L^2([0,1])$ and study their fixed matrix elements in the normalized shifted-Legendre basis. Three parameter regimes permit analytical control. At $r=5/2$, every fixed matrix element converges to $3δ_{kl}/5$, where $δ_{kl}$ is the Kronecker delta. At $r=16/5$, the attracting period-two orbit yields phase-resolved limits for the even and odd subsequences. At $r=4$, an exact Chebyshev-moment representation gives an $O(4^{-n})$ approach of every fixed matrix element at iteration $n$ to $δ_{kl}/2$. The case $r=37/10$ is treated only by a controlled finite numerical refinement study. Complementary finite diagnostics comprise matrix-element time dependence, a bifurcation-style plot, a time-averaged mean intensity, a normalized second-order intensity moment, and normalized scalar OTOC-type commutator correlation matrices. We also examine the separate finite-dimensional recursion $X_{k+1}=R X_k(I-X_k)R^\dagger$, with fixed matrix $R$, using diagonal and tridiagonal amplitude profiles. These operator-valued calculations are exploratory finite-time numerics. The analytical statements concern fixed matrix elements for the specified basis indices; they are distinct from the finite-resolution observations and do not imply operator-norm convergence. A regularized phase-space lift is included as a controlled visualization.
