Analytical perturbative approach to periodic orbits in the homogeneous quartic oscillator potential

Abstract

We present an analytical calculation of periodic orbits in the homogeneous quartic oscillator potential. Exploiting the properties of the periodic Lam{é} functions that describe the orbits bifurcated from the fundamental linear orbit in the vicinity of the bifurcation points, we use perturbation theory to obtain their evolution away from the bifurcation points. As an application, we derive an analytical semiclassical trace formula for the density of states in the separable case, using a uniform approximation for the pitchfork bifurcations occurring there, which allows for full semiclassical quantization. For the non-integrable situations, we show that the uniform contribution of the bifurcating period-one orbits to the coarse-grained density of states competes with that of the shortest isolated orbits, but decreases with increasing chaoticity parameter $\alpha$.

15 pages, LaTeX, 7 figures; revised and extended version, to appear in J. Phys. A final version 3; error in eq. (33) corrected and note added in print

References in corpus (1)

• Occurrence of periodic Lamé functions at bifurcations in chaotic Hamiltonian systems

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