Searcharxiv⌕ Search

arXiv subjects

Marko Robnik

Publications and source records attributed to Marko Robnik.

At least 55 records · Page 3Linked 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↗
arXiv2006

Energy evolution in time-dependent harmonic oscillator with arbitrary external forcing

The classical Hamiltonian system of time-dependent harmonic oscillator driven by the arbitrary external time-dependent force is considered. Exact analytical solution of the corresponding equations of motion is constructed in the framework of the technique (Robnik M, Romanovski V G, J. Phys. A: Math. Gen. {\bf 33} (2000) 5093) based on WKB approach. Energy evolution for the ensemble of uniformly distributed w.r.t. the canonical angle initial conditions on the initial invariant torus is studied. Exact expressions for the energy moments of arbitrary order taken at arbitrary time moment are analytically derived. Corresponding characteristic function is analytically constructed in the form of infinite series and numerically evaluated for certain values of the system parameters. Energy distribution function is numerically obtained in some particular cases. In the limit of small initial ensemble's energy the relevant formula for the energy distribution function is analytically derived.

nlin.SI↗
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↗
arXiv2002

Semiclassical analysis of Wigner functions

In this work we study the Wigner functions, which are the quantum analogues of the classical phase space density, and show how a full rigorous semiclassical scheme for all orders of \hbar can be constructed for them without referring to the actual coordinate space wavefunctions from which the Wigner functions are typically calculated. We find such a picture by a careful analysis around the stationary points of the main quantization equation, and apply this approach to the harmonic oscillator solving it for all orders of \hbar.

nlin.CD↗
arXiv2000

Topics in quantum chaos of generic systems

We review the main ideas and results in the stationary problems of quantum chaos in generic (mixed) systems, whose classical dynamics has regular (invariant tori) and chaotic regions coexisting in the phase space. First we discuss the universality classes of spectral fluctuations (GOE/GUE for ergodic systems, and Poissonian for integrable systems). We explain the problems in the calculation of the invariant (Liouville) measure of classically chaotic components, which has recently been studied by Robnik et al (1997) and by Prosen and Robnik (1998). Then we describe the Berry-Robnik (1984) picture, which is claimed to become exact in the strict semiclassical limit $\hbar\to 0$. However, at not sufficiently small values of $\hbar$ we see a crossover regime due to the localization properties of stationary quantum states where Brody-like behaviour with the fractional power law level repulsion is observed in the corresponding quantal energy spectra.

nlin.CD↗
arXiv2000

On WKB Series for the Radial Kepler Problem

We obtain the rigorous WKB expansion to all orders for the radial Kepler problem, using the residue calculus in evaluating the WKB quantization condition in terms of a complex contour integral in the complexified coordinate plane. The procedure yields the exact energy spectrum of this Schrödinger eigenvalue problem and thus resolves the controversies around the so-called "Langer correction". The problem is nontrivial also because there are only a few systems for which all orders of the WKB series can be calculated, yielding a convergent series whose sum is equal to the exact result, and thus sheds new light to similar and more difficult problems.

nlin.CD↗
arXiv2000

Regular and Irregular States in Generic Systems

In this work we present the results of a numerical and semiclassical analysis of high lying states in a Hamiltonian system, whose classical mechanics is of a generic, mixed type, where the energy surface is split into regions of regular and chaotic motion. As predicted by the principle of uniform semiclassical condensation (PUSC), when the effective $\hbar$ tends to 0, each state can be classified as regular or irregular. We were able to semiclassically reproduce individual regular states by the EBK torus quantization, for which we devise a new approach, while for the irregular ones we found the semiclassical prediction of their autocorrelation function, in a good agreement with numerics. We also looked at the low lying states to better understand the onset of semiclassical behaviour.

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↗
arXiv2000

On one-dimensional Schroedinger problems allowing polynomial solutions

We discuss the explicit construction of the Schroedinger equations admitting a representation through some family of general polynomials. Almost all solvable quantum potentials are shown to be generated by this approach. Some generalization has also been performed in higher-dimensional problems.

nlin.CD↗
arXiv2000

WKB corrections to the energy splitting in double well potentials

By using the WKB quantization we deduce an analytical formula for the energy splitting in a double--well potential which is the usual Landau formula with additional quantum corrections. Then we analyze the accuracy of our formula for the double square well potential, the inverted harmonic oscillator and the quartic potential.

nlin.CD↗
arXiv2000

Experimental study of generic billiards with microwave resonators

In this work we study the eigenstates and the energy spectra of a generic billiard system with the use of microwave resonators. This is possible due to the exact correspondence between the Schroedinger equation and the electric field equations of the lowest modes in thin microwave resonators. We obtain a good agreement between the numerical (exact) and experimental eigenstates, while the short range experimental spectral statistics show the expected Brody-like behaviour in this energy range, as opposed to the Berry-Robnik picture which is valid only in the semiclassical region of sufficiently small effective Planck's constant.

nlin.CD↗
arXiv2000

Study of Regular and Irregular States in Generic Systems

In this work we semiclassically analyzed the high lying eigenstates of a mixed type Hamiltonian system. For the regular states we employ the Einstein-Brillouin-Keller quantization, while for the chaotic states, following the principle of uniform semiclassical condensation, we obtain a prediction for their wavefunction autocorrelation function.

nlin.CD↗
arXiv2000

Accuracy of the WKB approximation: the case of general quartic potential

We analyse the accuracy of the approximate WKB quantization for the case of general one-dimensional quartic potential. In particular, we are interested in the validity of semiclassically predicted energy eigenvalues when approaching the limit $E\to \infty$, and in the accuracy of low lying energy levels below the potential barrier in the case of generally asymmetric double-well quartic potential. In the latter case, using the standard WKB quantization an unnatural localization of eigenstates due to the negligence of tunneling is implied and thus the validity of semiclassics is uncertain. In all computations the higher order corrections to the leading semiclassical approximation are included using the complex contour integration technique. We show that these corrections can improve accuracy of semiclassical approximation greatly by many orders of magnitude.

nlin.CD↗
arXiv2000

On the semiclassical expansion for 1-dim $x^N$ potentials

In the present paper we study the structure of the WKB series for the polynomial potential $V(x)=x^N$ ($N$ even). In particular, we obtain relatively simple recurrence formula of the coefficients $\s'_k$ of the semiclassical approximation and of the WKB terms for the energy eigenvalues.

nlin.CD↗
arXiv2000

Study of Spectral Statistics of Classically Integrable Systems

In this work we present the results of a study of spectral statistics for a classically integrable system, namely the rectangle billiard. We show that the spectral statistics are indeed Poissonian in the semiclassical limit for almost all such systems, the exceptions being the atypical rectangles with rational squared ratio of its sides, and of course the energy ranges larger than L_{\rm max}=\hbar / T_0$, where $T_0$ is the period of the shortest periodic orbit of the system, however $L_{\rm max} \to \infty$ when $E \to \infty$.

nlin.CD↗