Analytical perturbative approach to periodic orbits in the homogeneous quartic oscillator potential
نویسندگان
چکیده
We present an analytical calculation of periodic orbits in the homogeneous quartic oscillator potential. Exploiting the properties of the periodic Lamé functions that describe the orbits bifurcated from the fundamental linear orbit in the vicinity of the bifurcation points, we use standard perturbation theory to obtain their evolution away from the bifurcation points. As an application, we derive an analytical trace formula for the density of states in a separable case, including a uniform approximation for a pitchfork bifurcation of isolated orbits occurring there, leading to full semiclassical quantization.
منابع مشابه
Periodic Orbits and Scaling Laws for a Driven Damped Quartic Oscillator
In this paper we investigate the conditions under which periodic solutions of the nonlinear oscillator ẍ + x3 = 0 persist when the differential equation is perturbed so as to become ẍ + x3 + εx3 cos t + γẋ = 0. We conjecture that for any periodic orbit, characterized by its frequency ω, there exists a threshold for the damping coefficient γ, above which the orbit disappears, and that this thres...
متن کاملSelection Rules for Periodic Orbits and Scaling Laws for a Driven Damped Quartic Oscillator
In this paper we investigate the conditions under which periodic solutions of the nonlinear oscillator ẍ + x = 0 persist when the differential equation is perturbed so as to become ẍ + x + εx cos t+ γẋ = 0. For any frequency ω, there exists a threshold for the damping coefficient γ, above which there is no periodic orbit with period 2π/ω. We conjecture that this threshold is infinitesimal in th...
متن کاملBifurcation of Critical Periods from a Quartic Isochronous Center
This paper is focused on the bifurcation of critical periods from a quartic rigidly isochronous center under any small quartic homogeneous perturbations. By studying the number of zeros of the first several terms in the expansion of the period function in ε, it shows that under any small quartic homogeneous perturbations, up to orders 1 and 2 in ε, there are at most two critical periods bifurca...
متن کامل-symmetric quartic anharmonic oscillator and position-dependent mass in a perturbative approach
To lowest order of perturbation theory we show that an equivalence can be established between a PT -symmetric generalized quartic anharmonic oscillator model and a Hermitian position-dependent mass Hamiltonian h. An important feature of h is that it reveals a domain of couplings where the quartic potential could be attractive, vanishing or repulsive. We also determine the associated physical qu...
متن کاملA QUARTIC POTENTIAL FOR THE NUCLEONIC QUARKS
We assume that each valence quark in a nucleon is in a phenomenological modified harmonic oscillator potential of the form: ( l+yo) (ar +br+cr +dr ), where a, b, c and d are constants and ? is one of the Dirac matrices. Then by making use of a suitable ansatz, the Dirac equation has a very simple solution which is exact. We then have calculated the static properties of the nucleon in the ...
متن کامل