An integral semiclassical method for calculating the spectra for centrally symmetric potentials

A universal effective quantum number for centrally symmetric problems

“…The same value of φ m was obtained earlier using a duality between certain pairs of power-law potentials [15]. As we will see, χ d is a smooth monotonic function of d for all interesting model potentials.…”

“…The same value of φ m we have received earlier using a duality between certain pairs of power-law potentials [14].…”

“…along the vertical line φ = const. Each shell can contain D(l, d) (15) states; only this number D depends on d. When the shell (n r , l) is filled in, i.e. all states are occupied with particles, the filling of the next shell (n r , l ) begins with the value T > T nearest to T.…”

“…for any centrally symmetric potentials and any dimensionality of the problem by means of some generalization and new universal variations of the methods used earlier [13,14,16] in order to obtain T for d = 3. The coefficient φ is determined as some definite combination of the functionals N d [E; V ], which represent asymptotic estimates for the total number of the bound states in a given potential V (r) with the energies not exceeding E. These estimates are obtained without using any WKB method.…”

New approximate radial wave functions for power-law potentials

“…We calculate the values δe for the linear and quartic potentials with respect to the accurate numerical eigenenergies [12]. We compare our results with the recent results in the framework of an integral semiclassical method for calculating the spectra for spherically symmetric potentials [13]. In the case of linear potential with l = 0 our approach gives exact eigenfunctions and eigenenergies.…”