arXiv subjects

Maciej Janowicz

Publications and source records attributed to Maciej Janowicz.

6 recordsLinked to original sources

arXiv2026

Quantum logistic map considered as discrete-time Heisenberg equation

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 $3\delta_{kl}/5$, where $\delta_{kl}$ 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 $\delta_{kl}/2$. 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.

math.DS↗

arXiv2026

Bargmann Zeros as a Diagnostic of the Tunneling Transition in Double-Well Quantum Systems

Complex zeros of wavefunctions represented as entire functions in Bargmann--Fock space encode structural information about the underlying quantum state. Prior work employed zero galleries of randomly generated polynomial superpositions of Fock states as visual fingerprints suitable for classification. Here we examine whether Bargmann zeros of physically realized eigenstates of one-dimensional anharmonic and double-well Hamiltonians carry a recognizable signature of the tunneling transition in the symmetric double well. Ground and first-excited eigenstates are obtained from a variational ansatz consisting of a physically motivated symbolic envelope multiplied by a small flexible correction network, trained by Rayleigh--Ritz minimization of the finite-difference Hamiltonian expectation value and validated to reproduce energies to within $\sim 10^{-5}\,\mathrm{Ha}$. The resulting wavefunctions are projected onto the harmonic-oscillator basis and the complex zeros of the truncated Bargmann polynomial are located by numerical root-finding. For harmonic and quartic-anharmonic potentials the zeros show no preferred orientation. For double-well eigenstates, by contrast, the zeros condense onto the imaginary axis. A sweep of the barrier parameter $a$ from $0.5$ to $2.3$ reveals a continuous migration of zeros toward the imaginary axis, concurrent with the exponential collapse of the tunneling splitting $\Delta(a) = E_1 - E_0$ over $3.5$ decades. This condensation is traced to a sign-alternation pattern in the Fock-coefficient spectrum that is characteristic of bimodally localized wavefunctions. The complex zero set of the Bargmann-represented wavefunction thereby provides a compact, purely analytic diagnostic for the tunneling regime of one-dimensional double-well Hamiltonians, extending the random-polynomial zero-image framework to physical eigenstates.

quant-ph↗

arXiv2014

Homotopy analysis method for stochastic differential equations

The homotopy analysis method known from its successful applications to obtain quasi-analytical approximations of solutions of ordinary and partial differential equations is applied to stochastic differential equations with Gaussian stochastic forces and to the Fokker-Planck equations. Only the simplest non-trivial examples of such equations are considered, but such that they can almost immediately be translated to those which appear in the stochastic quantization of a nonlinear scalar field theory. It has been found that the homotopy analysis method yields excellent agreement with exact results (when the latter are available) and appears to be a very promising approach in the calculations related to quantum field theory and quantum statistical mechanics.

cond-mat.stat-mech↗

arXiv2013

Ground states and dynamics of a trapped charged particle in the magnetic field

A system of two charged particles in a harmonic trap with additional magnetic field is considered. The problem is reduced to a single-particle one in relative coordinates. The ground- and lowest excited-state energies and wave functions are found. The ground state exhibits non-zero expectation value of the velocity (kinetic momentum) and the probability current density does not vanish as well. When the ground state becomes degenerate the expectation value of velocity becomes discontinuous. The effects associated with turning on of the magnetic field are studied by solving the appropriate time-dependent Schrödinger equation. No substantial differences between abrupt (discontinuous in time) and continuous switching on have been observed. Evolution of a wave packet which is initially Gaussian is also investigated. The wave packet loses its Gaussian nature and, after sufficiently large time, a system of diffractive maxima and minima is built.

quant-ph↗

arXiv2013

Coherence and pattern formation in coupled logistic-map lattices

Three quantitative measures of the spatiotemporal behavior of the coupled map lattices: reduced density matrix, reduced wave function, and an analog of particle number, have been introduced. They provide a quantitative meaning to the concept of coherence which in the context of complex systems have been used rather intuitively. Their behavior suggests that the logistic coupled-map lattices approach the states which resemble the condensed states of systems of Bose particles. In addition, pattern formation in two-dimensional coupled map lattices based on the logistic mapping has been investigated with respect to the non-linear parameter, the diffusion constant and initial as well as boundary conditions.

nlin.CD↗

arXiv2012

Statistical linearizations for stochastically quantized fields

The statistical linearization method known in nonlinear mechanics and random vibrations theory has been applied to stochastically quantized fields in finite temperature. It has been shown that even in its simplest form the method yields convenient implicit equations for the self-energy, equivalent to the Dyson-Schwinger equations resulting from the summation of infinite number of perturbative diagrams. Three examples have been provided: the quantum anharmonic oscillator, the scalar $ϕ^{4}$ theory in three spatial dimension, and the Bose-Hubbard model. The Ramanujan summation has been used to deal with divergent integrals and series.

hep-th↗