The anharmonic oscillator problem: a new algebraic solution

Abstract

The second-order (one-dimensional or radial) differential Schrodinger equation with the potential V(r)= mu r2+ nu r4 may be re-interpreted as a difference equation of the fourth order (indeed, the Hamiltonian is a pentadiagonal matrix in the standard harmonic oscillator basis mod n), n=0, 1, . . .). Thus, the authors construct its four independent general solutions by purely algebraic means, via expansions in powers of (n+1)-14/. Next, preserving the analogy between difference and differential equations, the physical wavefunctions (n mod psi ) and their energies are determined by the matching of the 'Jost' and 'regular' solutions. Finally, using the simplest matching conditions 'at the origin', the resulting numerical algorithm is demonstrated to be both stable and quickly convergent.