Characterizing weak chaos in nonintegrable Hamiltonian systems: the fundamental role of stickiness and initial conditions

Weak chaos in high-dimensional conservative systems can be characterized through sticky effect induced by invariant structures on chaotic trajectories. Suitable quantities for this characterization are the higher cummulants of the finite time Lyapunov exponents (FTLEs) distribution. They gather the {\it whole} phase space relevant dynamics in {\it one} quantity and give informations about ordered and random states. This is analyzed here for discrete Hamiltonian systems with local and global couplings. It is also shown that FTLEs plotted {\it versus} initial condition (IC) and the nonlinear parameter is essential to understand the fundamental role of ICs in the dynamics of weakly chaotic Hamiltonian systems.Comment: 7 pages, 6 figures, submitted for publicatio

Position as an independent variable and the emergence of the 1/21/2-time fractional derivative in quantum mechanics

Using the position as an independent variable, and time as the dependent variable, we derive the function ${\cal P}^{(\pm)}$, which generates the space evolution under the potential ${\cal V}(q)$ and Hamiltonian ${\cal H}$. Canonically conjugated variables are the time and minus the Hamiltonian. While the classical dynamics do not change, the corresponding quantum operator naturally leads to a $1/2-$fractional time evolution, consistent with a recently proposed spacetime symmetric formalism of quantum mechanics. Using Dirac's procedure, separation of variables is possible, and while the coupled position-independent Dirac equations depend on the $1/2$-fractional derivative, the coupled time-independent Dirac equations (TIDE) lead to positive and negative shifts in the potential, proportional to the force. Both equations couple the ($\pm$) solutions of ${\cal \hat P}^{(\pm)}$ and the kinetic energy ${\cal K}_0$ is the coupling strength. We obtain a pair of coupled states for systems with finite forces. The potential shifts for the harmonic oscillator (HO) are $\pm\hbar\omega/2$, and the corresponding pair of states are coupled for ${\cal K}_0\ne 0$. No time evolution is present for ${\cal K}_0=0$, and the ground state with energy $\hbar\omega/2$ is stable. For ${\cal K}_0>0$, the ground state becomes coupled to the state with energy $-\hbar\omega/2$, and \textit{this coupling} allows to describe higher excited states. Energy quantization of the HO leads to quantization of ${\cal K}_0=k\hbar\omega$ ($k=1,2,\ldots$). For the one-dimensional Hydrogen atom, the potential shifts become imaginary and position-dependent. Decoupled case ${\cal K}_0=0$ leads to plane-waves-like solutions at the threshold. Above the threshold, we obtain a plane-wave-like solution, and for the bounded states the wave-function becomes similar to the exact solutions but squeezed closer to the nucleus.Comment: 11 pages, 6 figure

Fractional integrodifferential equations and (anti-)hermiticity of time in a spacetime-symmetric extension of nonrelativistic Quantum Mechanics

Time continues to be an intriguing physical property in the modern era. On the one hand, we have the Classical and Relativistic notion of time, where space and time have the same hierarchy, which is essential in describing events in spacetime. On the other hand, in Quantum Mechanics, time appears as a classical parameter, meaning that it does not have an uncertainty relation with its canonical conjugate. In this work, we use a recent proposed spacetime-symmetric formalism~\href{https://doi.org/10.1103/PhysRevA.95.032133}{[Phys.~Rev.~A {\bf 95}, 032133 (2017)]} that tries to solve the unbalance in nonrelativistic Quantum Mechanics by extending the usual Hilbert space. The time parameter $t$ and the position operator $\hat{X}$ in one subspace, and the position parameter $x$ and time operator $\mathbb{T}$ in the other subspace. Time as an operator is better suitable for describing tunnelling processes. We then solve the novel $1/2$-fractional integrodifferential equation for a particle subjected to strong and weak potential limits and obtain an analytical expression for the tunnelling time through a rectangular barrier. We compare to previous works, obtaining pure imaginary times for energies below the barrier and a fast-decaying imaginary part for energies above the barrier, indicating the anti-hermiticity of the time operator for tunnelling times. We also show that the expected time of arrival in the tunnelling problem has the form of an energy average of the classical times of arrival plus a quantum contribution.Comment: 11 pages, 3 figure