Orthogonal polynomial projection quantization: a new Hill determinant method

Accurate energy eigenvalues are obtained by simply projecting the unknown bound state wave function on, essentially, arbitrary sets of orthogonal polynomials, and setting a subset of these to zero. The projection integrals are represented in terms of the power moments of the wave function, obtained recursively by transforming Schrodinger's equation into a moment equation. Because unbounded wave functions do not have power moments, all solutions are guaranteed to be L-2, resulting in a more robust, rapidly converging and stable method when compared with configuration space Hill determinant methods. More importantly, our approach permits the use of arbitrary, nonanalytic, positive reference functions, including those that manifest the true asymptotic behavior of the discrete states. These advantages are not usually possible with the standard Hill approach. The formulation presented here can be applied to any problem for which Schrodinger's equation can be transformed into a moment equation. In this regard it is related to the L-2 quantization prescription developed by Tymczak et al (1998 Phys. Rev. Lett. 80 3674; 1998 Phys. Rev. A 58 2708) corresponding to the Hill determinant method in momentum space, and to the eigenvalue moment method (Handy and Bessis 1985 Phys. Rev. Lett. 55 931; Handy et al 1988 Phys. Rev. Lett. 60 253), the first use of semidefinite programming related analysis in quantum physics. The latter method can be used to generate the ground state, whose orthogonal polynomials can serve to generate even more rapidly converging estimates for the excited discrete state energies. We demonstrate the power of this new approach on the sextic and quartic anharmonic oscillators, as well as on recently studied one- and two-dimensional pseudo-Hermitian Hamiltonians.