Time-dependent Hamiltonian Systems Research Articles

Power-law decay of the fraction of the mixed eigenstates in kicked top model with mixed-type classical phase space.

The properties of mixed eigenstates in a generic quantum system with a classical counterpart that has mixed-type phase space, although important to understand several fundamental questions that arise in both theoretical and experimental studies, are still not clear. Here, following a recent work [Č. Lozej, D. Lukman, and M. Robnik, Phys. Rev. E 106, 054203 (2022)2470-004510.1103/PhysRevE.106.054203], we perform an analysis of the features of mixed eigenstates in a time-dependent Hamiltonian system, the celebrated kicked top model. As a paradigmatic model for studying quantum chaos, the kicked top model is known to exhibit both classical and quantum chaos. The types of eigenstates are identified by means of the phase-space overlap index, which is defined as the overlap of the Husimi function with regular and chaotic regions in classical phase space. We show that the mixed eigenstates appear due to various tunneling precesses between different phase-space structures, while the regular and chaotic eigenstates are, respectively, associated with invariant tori and chaotic components in phase space. We examine how the probability distribution of the phase-space overlap index evolves with increasing system size for different kicking strengths. In particular, we find that the relative fraction of mixed states exhibits a power-law decay as the system size increases, indicating that only purely regular and chaotic eigenstates are left in the strict semiclassical limit. We thus provide further verification of the principle of uniform semiclassical condensation of Husimi functions and confirm the correctness of the Berry-Robnik picture.

The vicinity of Earth–Moon [formula omitted] and [formula omitted] in the Hill restricted 4-body problem

The Hill restricted 4-body problem (HR4BP) is a periodic time-dependent Hamiltonian dynamical system that is both a generalization of the circular restricted 3-body problem (CR3BP) and Hill’s problem. In this work, the HR4BP is used to model the motion of an infinitesimal particle under the mutual gravitation of the Sun, Earth, and Moon. In the HR4BP, the equilibria L1 and L2 of the CR3BP are replaced by π-periodic orbits, the so-called EM L1,2 dynamical equivalents. We compute center manifold reductions about EM L1,2 to obtain a local overview of the dynamics in their vicinity. In this way, we show the existence of families of 2- and 3-dimensional quasi-periodic invariant tori around EM L1,2. Additionally, we exploit the first integral constructed in the center manifold reduction to compute hyperbolic invariant manifolds. We draw comparisons to the bicircular and quasi-bicircular restricted 4-body problems.

An Introduction to PT-Symmetric Quantum Mechanics-Time-Dependent Systems

I will provide a pedagogical introduction to non-Hermitian quantum systems that are PT-symmetric, that is they are left invariant under a simultaneous parity transformation (P) and time-reversal (T). I will explain how generalised versions of this antilinear symmetry can be utilised to explain that these type of systems possess real eigenvalue spectra in parts of their parameter spaces and how to set up a consistent quantum mechanical framework for them that enables a unitary time-evolution. In the second part I will explain how to extend this framework to explicitly time-dependent Hamiltonian systems and report in particular on recent progress made in this context. I will explain how to construct the essential key quantity in this framework, the time-dependent Dyson map and metric and solutions to the time-dependent Schrödinger equation, in an algebraic fashion, using time-dependent Darboux transformations, utilising Lewis-Riesenfeld invariants, point transformations and some approximation methods. I comment on the ambiguities of this metric and demonstrate that this can even lead to infinite series of metric operators. I conclude with some applications to PT-symmetrically coupled oscillators, demonstrate the equivalence of the time-dependent double wells and unstable anharmonic oscillators and show how the unphysical PT-symmetrically broken regions in the parameter space for the time-independent theory becomes physical in the explicitly time-dependent systems. I discuss how this leads to a prolongation of the otherwise rapidly decaying von Neumann entropy. The so-called sudden death of the entropy is stopped at a finite value.1

Exact solutions and superposition rules for Hamiltonian systems generalizing time-dependent SIS epidemic models with stochastic fluctuations

<abstract><p>Using the theory of Lie-Hamilton systems, formal generalized time-dependent Hamiltonian systems that extend a recently proposed SIS epidemic model with a variable infection rate are considered. It is shown that, independently on the particular interpretation of the time-dependent coefficients, these systems generally admit an exact solution, up to the case of the maximal extension within the classification of Lie-Hamilton systems, for which a superposition rule is constructed. The method provides the algebraic frame to which any SIS epidemic model that preserves the above-mentioned properties is subjected. In particular, we obtain exact solutions for generalized SIS Hamiltonian models based on the book and oscillator algebras, denoted by $ \mathfrak{b}_2 $ and $ \mathfrak{h}_4 $, respectively. The last generalization corresponds to an SIS system possessing the so-called two-photon algebra symmetry $ \mathfrak{h}_6 $, according to the embedding chain $ \mathfrak{b}_2\subset \mathfrak{h}_4\subset \mathfrak{h}_6 $, for which an exact solution cannot generally be found but a nonlinear superposition rule is explicitly given.</p></abstract>

Phase Space Reaction Dynamics Associated with an Index-2 Saddle Point for Time-Dependent Hamiltonian Systems

We analyze benchmark models for reaction dynamics associated with a time-dependent index-2 saddle point. The influence of index-2 saddle points on chemical reaction dynamics has received a great deal of attention in recent years and we extend this work in a new and important direction. Our model allows us to incorporate time dependence of a general form, subject to an exponential growth restriction. Under these conditions, we analytically compute the time-dependent normally hyperbolic invariant manifold and its time-dependent stable and unstable manifolds. These time-dependent phase space structures allow us to quantitatively determine the effect of the time dependence on reacting and nonreacting trajectories.

A Variational Formulation of Accelerated Optimization on Riemannian Manifolds

It was shown recently by Su et al. (2016) that Nesterov's accelerated gradient method for minimizing a smooth convex function $f$ can be thought of as the time discretization of a second-order ODE, and that $f(x(t))$ converges to its optimal value at a rate of $\mathcal{O}(1/t^2)$ along any trajectory $x(t)$ of this ODE. A variational formulation was introduced in Wibisono et al. (2016) which allowed for accelerated convergence at a rate of $\mathcal{O}(1/t^p)$, for arbitrary $p>0$, in normed vector spaces. This framework was exploited in Duruisseaux et al. (2020) to design efficient explicit algorithms for symplectic accelerated optimization. In Alimisis et al. (2020), a second-order ODE was proposed as the continuous-time limit of a Riemannian accelerated algorithm, and it was shown that the objective function $f(x(t))$ converges to its optimal value at a rate of $\mathcal{O}(1/t^2)$ along solutions of this ODE. In this paper, we show that on Riemannian manifolds, the convergence rate of $f(x(t))$ to its optimal value can also be accelerated to an arbitrary convergence rate $\mathcal{O}(1/t^p)$, by considering a family of time-dependent Bregman Lagrangian and Hamiltonian systems on Riemannian manifolds. This generalizes the results of Wibisono et al. (2016) to Riemannian manifolds and also provides a variational framework for accelerated optimization on Riemannian manifolds. An approach based on the time-invariance property of the family of Bregman Lagrangians and Hamiltonians was used to construct very efficient optimization algorithms in Duruisseaux et al. (2020), and we establish a similar time-invariance property in the Riemannian setting. One expects that a geometric numerical integrator that is time-adaptive, symplectic, and Riemannian manifold preserving will yield a class of promising optimization algorithms on manifolds.

Solutions of The Hamilton Equations for Time-Dependent Hamiltonian Systems by Means of Solvable Lie Algebras of Symmetries

In this paper we consider Hamiltonian systems where the Hamiltonian function may depend explicitly on time and we show that some results existing for Hamiltonian systems in the time-independent framework can be extended to the time-dependent case. In particular, we show that if we have a sufficient number of functionally independent (possible time-dependent) constants of motion that generate a solvable Lie algebra under the Poisson bracket , then the solutions of the Hamilton equations of motion can be found by quadratures . For this purpose, we prove that the constants of motion are related to symmetries of the Hamiltonian vector field X H that define the dynamics of the system, and if such constants of motion generate a solvable Lie algebra then we can find a solvable Lie algebra of symmetries of X H , which permit applying a reduction process for systems of differential equations up to the point of solving it by integrating some known functions and doing some algebraic operations .

Dynamical invariants for time-dependent real and complex Hamiltonian systems

The Struckmeier and Riedel (SR) approach is extended in real space to isolate dynamical invariants for one- and two-dimensional time-dependent Hamiltonian systems. We further develop the SR-formalism in zz̄ complex phase space characterized by z = x + iy and z̄=x−iy and construct invariants for some physical systems. The obtained quadratic invariants contain a function f2(t), which is a solution of a linear third-order differential equation. We further explore this approach into extended complex phase space defined by x = x1 + ip2 and p = p1 + ix2 to construct a quadratic invariant for a time-dependent quadratic potential. The derived invariants may be of interest in the realm of numerical simulations of explicitly time-dependent Hamiltonian systems.

Analysis of Novel Oscillations of Quantized Mechanical Energy in Mass-Accreting Nano-Oscillator Systems

Quantum characteristics of a mass-accreting oscillator are investigated using the invariant operator theory, which is a rigorous mathematical tool for unfolding quantum theory for time-dependent Hamiltonian systems. In particular, the quantum energy of the system is analyzed in detail and compared to the classical one. We focus on two particular cases; one is a linearly mass-accreting oscillator and the other is an exponentially mass-accreting one. It is confirmed that the quantum energy is in agreement with the classical one in the limit ℏ→0. We showed that not only the classical but also the quantum energy oscillates with time. It is carefully analyzed why the energy oscillates with time, and a reasonable explanation for that outcome is given.

Exactly solvable time-dependent non-Hermitian quantum systems from point transformations

We demonstrate that complex point transformations can be used to construct non-Hermitian first integrals, time-dependent Dyson maps and metric operators for non-Hermitian quantum systems. Initially we identify a point transformation as a map from an exactly solvable time-independent system to an explicitly time-dependent non-Hermitian Hamiltonian system. Subsequently we employ the point transformation to construct the non-Hermitian time-dependent invariant for the latter system. Exploiting the fact that this invariant is pseudo-Hermitian, we construct a corresponding Dyson map as the adjoint action from a non-Hermitian to a Hermitian invariant, thus obtaining solutions to the time-dependent Dyson and time-dependent quasi-Hermiticity equation together with solutions to the corresponding time-dependent Schrödinger equation.

Dirac’s Formalism for Time-Dependent Hamiltonian Systems in the Extended Phase Space

Dirac’s formalism for constrained systems is applied to the analysis of time-dependent Hamiltonians in the extended phase space. We show that the Lewis invariant is a reparametrization invariant, and we calculate the Feynman propagator using the extended phase space description. We show that the Feynman propagator’s quantum phase is given by the boundary term of the canonical transformation of the extended phase space. We propose a new canonical transformation within the extended phase space that leads to a Lewis invariant generalization, and we sketch some possible applications.

Nonexistence of invariant tori transverse to foliations: An application of converse KAM theory.

Invariant manifolds are of fundamental importance to the qualitative understanding of dynamical systems. In this work, we explore and extend MacKay's converse Kolmogorov-Arnol'd-Moser condition to obtain a sufficient condition for the nonexistence of invariant surfaces that are transverse to a chosen 1D foliation. We show how useful foliations can be constructed from approximate integrals of the system. This theory is implemented numerically for two models: a particle in a two-wave potential and a Beltrami flow studied by Zaslavsky (Q-flows). These are both 3D volume-preserving flows, and they exemplify the dynamics seen in time-dependent Hamiltonian systems and incompressible fluids, respectively. Through both numerical and theoretical considerations, it is revealed how to choose foliations that capture the nonexistence of invariant tori with varying homologies.

Perturbation Theory for Time-Dependent Quantum Systems Involving Complex Potentials

We explore how to apply perturbation theory to complicated time-dependent Hamiltonian systems that involve complex potentials. To do this, we introduce a generalized time-dependent oscillator to which the complex potentials are connected through a weak coupling strength. We regard the complex potentials in the Hamiltonian as the perturbed terms. Quantum characteristics of the system, such as wave functions and expectation values of the Hamiltonian, are investigated on the basis of the perturbation theory. We apply our theory to particular systems with explicit choices of time-dependent parameters. Through such applications, the time behavior of the quantum wave packets and the spectrum of expectation values of the Hamiltonian are analyzed in detail. We confirm that the imaginary parts of expectation values of the Hamiltonian are not zero but very small, while the real parts deviate slightly from those of the unperturbed system.

LONG TIME BEHAVIOR OF THE SOLUTIONS OF NLW ON THE -DIMENSIONAL TORUS

We consider the nonlinear wave equation (NLW) on the $d$ -dimensional torus $\mathbb{T}^{d}$ with a smooth nonlinearity of order at least 2 at the origin. We prove that, for almost any mass, small and smooth solutions of high Sobolev indices are stable up to arbitrary long times with respect to the size of the initial data. To prove this result, we use a normal form transformation decomposing the dynamics into low and high frequencies with weak interactions. While the low part of the dynamics can be put under classical Birkhoff normal form, the high modes evolve according to a time-dependent linear Hamiltonian system. We then control the global dynamics by using polynomial growth estimates for high modes and the preservation of Sobolev norms for the low modes. Our general strategy applies to any semilinear Hamiltonian Partial Differential Equations (PDEs) whose linear frequencies satisfy a very general nonresonance condition. The (NLW) equation on $\mathbb{T}^{d}$ is a good example since the standard Birkhoff normal form applies only when $d=1$ while our strategy applies in any dimension.

Jumps of Energy Near a Homoclinic Set of a Slowly Time Dependent Hamiltonian System

We consider a Hamiltonian system depending on a parameter which slowly changes with rate e ≪ 1. If trajectories of the frozen autonomous system are periodic, then the system has adiabatic invariant which changes much slower than energy. For a system with 1 degree of freedom and a figure 8 separatrix, Anatoly Neishtadt [18] showed that for trajectories crossing the separatrix, the adiabatic invariant, and hence the energy, have quasirandom jumps of order e. We prove a partial analog of Neishtadt’s result for a system with n degrees of freedom such that the frozen system has a hyperbolic equilibrium possessing several homoclinic orbits. We construct trajectories staying near the homoclinic set with energy having jumps of order e at time intervals of order ∣ln e∣, so the energy may grow with rate e/∣ln e∣. Away from the homoclinic set faster energy growth is possible: if the frozen system has chaotic behavior, Gelfreich and Turaev [16] constructed trajectories with energy growth rate of order e.

Analysis of the geometric phase for a nanowire-bridged superconducting Fabry-Perot resonator

The geometric phases of a nanowire-bridged superconducting Fabry-Perot resonator subjected to a microwave transmission have been investigated through its modelling into a RLC-circuit. Because the Hamiltonian of the system is a somewhat complicated form, special mathematical techniques, such as the invariant operator method and the unitary transformation approach, have been adopted in order to treat the system; These methods are very useful for managing complicated time-dependent Hamiltonian systems. We have rigorously evaluated the analytical geometric phases in both the Fock and coherent states. Typically, the geometric phases oscillate and the amplitude of such oscillations tend to grow over time. The influence of parameters of the system on the geometric phases has been analyzed in detail through the relevant illustrations. From our research, the concept of geometric phases and associated quantum mechanical characters of the system has been clarified. Our investigation for the geometric phases is useful for understanding topological features of the system, that take place through the evolution of the wave functions.

Simply-Laced Quantum Connections Generalising KZ

We construct a new family of flat connections generalising the KZ connection, the Casimir connection and the dynamical connection. These new connections are attached to simply-laced graphs, and are obtained via quantisation of time-dependent Hamiltonian systems controlling the isomonodromic deformations of meromorphic connections on the sphere.

Noncommutative Painlevé Equations and Systems of Calogero Type

All Painlev\'e equations can be written as a time-dependent Hamiltonian system, and as such they admit a natural generalization to the case of several particles with an interaction of Calogero type (rational, trigonometric or elliptic). Recently, these systems of interacting particles have been proved to be relevant in the study of $\beta$-models. An almost two decade old open question by Takasaki asks whether these multi-particle systems can be understood as isomonodromic equations, thus extending the Painlev\'e correspondence. In this paper we answer in the affirmative by displaying explicitly suitable isomonodromic Lax pair formulations. As an application of the isomonodromic representation we provide a construction based on discrete Schlesinger transforms, to produce solutions for these systems for special values of the coupling constants, starting from uncoupled ones; the method is illustrated for the case of the second Painlev\'e equation.

On time-dependent Hamiltonian realizations of planar and nonplanar systems

In this paper, we elucidate the key role played by the cosymplectic geometry in the theory of time dependent Hamiltonian systems. In particular, we generalize the cosymplectic structures to time-dependent Nambu-Poisson Hamiltonian systems and corresponding Jacobi's last multiplier for 3D systems. We illustrate our constructions with various examples.

Stokes Phenomenon and Confluence in Non-autonomous Hamiltonian Systems

This article studies a confluence of a pair of regular singular points to an irregular one in a generic family of time-dependent Hamiltonian systems in dimension 2. This is a general setting for the understanding of the degeneration of the sixth Painlevé equation to the fifth one. The main result is a theorem of sectoral normalization of the family to an integrable formal normal form, through which is explained the relation between the local monodromy operators at the two regular singularities and the non-linear Stokes phenomenon at the irregular singularity of the limit system. The problem of analytic classification is also addressed.