1/n-expansion for asymptotic coefficients of radial wave functions in quantum mechanics

Using the 1/n-expansion, we obtain analytic formulae for the bound state radial wave functions, including its asymptotic coefficients at r --> 0 and r --> infinity, for an arbitrary smooth potential V(r). The formulae are asymptotically exact in the limit n( )r--> infinity (n = n(r) + l + 1 is the principal quantum number and the expansion parameter is 1/n). Comparison with exact solutions and numerical calculations for the power-law and short-range potentials show that the applicability region of these formulae is usually prolonged up to small quantum numbers, n similar to 1. With growing n(r), the accuracy of the formulae decreases, but the WKB method becomes applicable in this case.