Quantum logistic map considered as discrete-time Heisenberg equation

Abstract

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.

Disclosure

“ve AI tools were used in two distinct capacities. First, AI-assisted code generation (ChatGPT Work and Codex) was used for creation of Python routines; all final implementation was written, inspected, and verified by the authors. Second, a large language model (ChatGPT Work) was used to assist with prose editing and language refinement at the manuscript stage. The authors reviewed and took responsibility for all content thus produced. No AI tool is listed as an author; all authors accept full re”

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Rewriting existing author-written text

Multiplier

4

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2026-08-23T19:01:25.217319Z

Structural counts