arXiv subjects

Valery G. Romanovski

Publications and source records attributed to Valery G. Romanovski.

23 records · Page 2Linked to original sources

arXiv2006

Exact energy distribution function in time-dependent harmonic oscillator

Following a recent work by Robnik and Romanovski (J.Phys.A: Math.Gen. {\bf 39} (2006) L35, Open Syst. & Infor. Dyn. {\bf 13} (2006) 197-222) we derive the explicit formula for the universal distribution function of the final energies in a time-dependent 1D harmonic oscillator, whose functional form does not depend on the details of the frequency $ω(t)$, and is closely related to the conservation of the adiabatic invariant. The normalized distribution function is $P(x) = π^{-1} (2μ^2 - x^2)^{-{1/2}}$, where $x=E_1- \bar{E_1}$, $E_1$ is the final energy, $\bar{E_1}$ is its average value, and $μ^2$ is the variance of $E_1$. $\bar{E_1}$ and $μ^2$ can be calculated exactly using the WKB approach to all orders.

nlin.SI↗

arXiv2006

Energy evolution in time-dependent harmonic oscillator

The theory of adiabatic invariants has a long history, and very important implications and applications in many different branches of physics, classically and quantally, but is rarely founded on rigorous results. Here we treat the general time-dependent one-dimensional harmonic oscillator, whose Newton equation $\ddot{q} + ω^2(t) q=0$ cannot be solved in general. We follow the time-evolution of an initial ensemble of phase points with sharply defined energy $E_0$ at time $t=0$ and calculate rigorously the distribution of energy $E_1$ after time $t=T$, which is fully (all moments, including the variance $μ^2$) determined by the first moment $\bar{E_1}$. For example, $μ^2 = E_0^2 [(\bar{E_1}/E_0)^2 - (ω(T)/ω(0))^2]/2$, and all higher even moments are powers of $μ^2$, whilst the odd ones vanish identically. This distribution function does not depend on any further details of the function $ω(t)$ and is in this sense universal. In ideal adiabaticity $\bar{E_1} = ω(T) E_0/ω(0)$, and the variance $μ^2$ is zero, whilst for finite $T$ we calculate $\bar{E_1}$, and $μ^2$ for the general case using exact WKB-theory to all orders. We prove that if $ω(t)$ is of class ${\cal C}^{m}$ (all derivatives up to and including the order $m$ are continuous) $μ\propto T^{-(m+1)}$, whilst for class ${\cal C}^{\infty}$ it is known to be exponential $μ\propto \exp (-αT)$.

nlin.CD↗

arXiv2005

Exact Analysis of the Adiabatic Invariants in Time-Dependent Harmonic Oscillator

The theory of adiabatic invariants has a long history and important applications in physics but is rarely rigorous. Here we treat exactly the general time-dependent 1-D harmonic oscillator, $\ddot{q} + ω^2(t) q=0$ which cannot be solved in general. We follow the time-evolution of an initial ensemble of phase points with sharply defined energy $E_0$ and calculate rigorously the distribution of energy $E_1$ after time $T$, and all its moments, especially its average value $\bar{E_1}$ and variance $μ^2$. Using our exact WKB-theory to all orders we get the exact result for the leading asymptotic behaviour of $μ^2$.

nlin.CD↗

arXiv2000

On Urabe's criteria of isochronicity

We give a short proof of Urabe's criteria for the isochronicity of periodical solutions of the equation $\ddot{x}+g(x)=0$. We show that apart from the harmonic oscillator there exists a large family of isochronous potentials which must all be non-polynomial and not symmetric (an even function of the coordinate x).

nlin.CD↗

arXiv2000

Some properties of WKB series

We investigate some properties of the WKB series for arbitrary analytic potentials and then specifically for potentials $x^N$ ($N$ even), where more explicit formulae for the WKB terms are derived. Our main new results are: (i) We find the explicit functional form for the general WKB terms $σ_k'$, where one has only to solve a general recursion relation for the rational coefficients. (ii) We give a systematic algorithm for a dramatic simplification of the integrated WKB terms $\oint σ_k'dx$ that enter the energy eigenvalue equation. (iii) We derive almost explicit formulae for the WKB terms for the energy eigenvalues of the homogeneous power law potentials $V(x) = x^N$, where $N$ is even. In particular, we obtain effective algorithms to compute and reduce the terms of these series.

nlin.CD↗