Semi-Classical quantization nonmanifestly using the method of harmonic balance

Semiclassical treatment of logarithmic perturbation theory

“…To our knowledge, there are only a few procedures for computing the coefficients E i (ω, n). They involve, in particular, applying the methods of the comparison equation [21] and complex "sprout" [22]; an analytic continuation in the h -plane [23]; various approaches within the framework of the WKB approximation [15][16][17][18][19]; quantization using the methods of classical mechanics [17,24]; and, lastly, expansions in the h1/2 -series [25]. However, all of these methods have some disadvantages.…”

A Recursion Technique for Deriving Renormalized Perturbation Expansions for One-Dimensional Anharmonic Oscillator

“…A few procedures are elaborated for computing the coefficients E i (ω, n). They involve, in particular, applying the methods of the comparison equation 21 and complex 'sprout' 22 ; an analytic continuation in the h-plane 23 ; various approaches within the framework of the WKB-approximation 24−28 ; quantization using the methods of classical mechanics 26,29 ; and, lastly, expansions in the h1/2 -series 30 .…”

“…Though proposed technique is easily applied to the mass renormalization we do consider only the case of renormalization of a frequency as being more physically motivated. For this purpose it is enough to think of the harmonic oscillator frequency incoming in the potential as a function of Plank's constant variable, with subsequent its expansion in an h-series 29 . However, for later use it is more convenient to take…”