Radial anharmonic oscillator: Perturbation theory, new semiclassical expansion, approximating eigenfunctions. I. Generalities, cubic anharmonicity case

For the general D-dimensional radial anharmonic oscillator with potential V(r) = 1/g(2)(V) over cap (gr) the perturbation theory (PT) in powers of coupling constant g (weak coupling regime) and in inverse, fractional powers of g (strong coupling regime) is developed constructively in r-space and in (gr)-space, respectively. The Riccati-Bloch (RB) equation and generalized Bloch (GB) equation are introduced as ones which govern dynamics in coordinate r-space and in (gr)-space, respectively, exploring the logarithmic derivative of wave function y. It is shown that PT in powers of g developed in RB equation leads to Taylor expansion of y at small r while being developed in GB equation leads to a new form of semiclassical expansion at large (gr): it coincides with loop expansion in path integral formalism. In complementary way PT for large g developed in RB equation leads to an expansion of y at large r and developed in GB equation leads to an expansion at small (gr). Interpolating all four expansions for y leads to a compact function (called the Approximant), which should uniformly approximate the exact eigenfunction at r is an element of [0, infinity) for any coupling constant g >= 0 and dimension D > 0. As a concrete application, the low-lying states of the cubic anharmonic oscillator V = r(2) + gr(3) are considered. 3 free parameters of the Approximant are fixed by taking it as a trial function in variational calculus. It is shown that the relative deviation of the Approximant from the exact ground state eigenfunction is less than or similar to 10(-4) for r is an element of [0, infinity) for coupling constant g >= 0 and dimension D = 1, 2, .... In turn, the variational energies of the low-lying states are obtained with unprecedented accuracy 7-8 s.d. for g >= 0 and D = 1, 2, ....