M Brack et al 2003 J. Phys. A: Math. Gen. 36 1095 doi:10.1088/0305-4470/36/4/317
M Brack1, S N Fedotkin1,2, A G Magner1,2 and M Mehta1,3
Show affiliationsWe 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 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 α.
03.65.Sq Semiclassical theories and applications
03.65.Db Functional analytical methods
37G10 Bifurcations of singular points
81Q10 Selfadjoint operator theory in quantum theory, including spectral analysis
81Q20 Semiclassical techniques including WKB and Maslov methods
Issue 4 (31 January 2003)
Received 30 July 2002, in final form 3 December 2002
Published 15 January 2003
M Brack et al 2003 J. Phys. A: Math. Gen. 36 1095
X Xiong et al 2003 Metrologia 40 S89
Fabrizio Lillo et al 2008 New J. Phys. 10 043019
Y Tsuchiva 1987 J. Phys. C: Solid State Phys. 20 1209
Ayhan Ozdemir 2004 Meas. Sci. Technol. 15 1316
J A Champion 1964 Br. J. Appl. Phys. 15 633
D F T Mullamphy et al 2007 J. Phys. B: At. Mol. Opt. Phys. 40 1141
Patrick L Chow et al 2005 Phys. Med. Biol. 50 1837
E Ascher and D Gay 1985 J. Phys. A: Math. Gen. 18 397
W Sucksmith 1957 Br. J. Appl. Phys. 8 S24