Orthogonal Polynomial Projection Quantization: A New Hill Determinant Formulation

We present a new formulation\footnote{C. R. Handy and D. Vrinceanu, to appear J. Phys A: Math. Theor. (2013). } (OPPQ) of the configuration space Hill determinant (HD) approach,\footnote{K. Banerjee, Proc. R. Soc. Lond. A 368 155 (1979).} and its momentum space counterpart (MRF),\footnote{C. J. Tymczak, G. S. Japaridze, C. R. Handy, and Xiao-Qian Wang, Phys. Rev. Lett. 80, 3674 (1998).} that has non of the instabilities of the former,\footnote{A. Hautot, Phys. Rev. D 33, 437 (1986).} nor the limitations of both. Let $\Psi $(x) $= \quad \Sigma_{n}$ a$_{n}$ P$_{n}$(x) R(x), where the P$_{n}$ `s are the orthogonal polynomials for a given \textit{reference function}, R(x) \textgreater 0 . If the system admits a linear recursive \textit{moment equation} representation, the a$_{n}$`s become a finite sum with respect to the moments $\mu_{p} \quad = \quad \smallint $ x$^{p} \quad \Psi $(x). Constraining a$_{N} \quad =$ 0,\textellipsis ,a$_{N-ms\, }=$ 0 gives impressive results for the discrete state energies, surpassing MRF. Contrary to HD, OPPQ is an L$^{2}$ formulation in which R(x): (i) does not have to be analytic; and (ii) can be adapted to the asymptotic form of $\Psi $. It has been applied to 1D and 2D anharmonic potentials, including pseudo hermitian systems, as well as the difficult two dimensional dipole problem for modeling edge structures in nanomaterials.\footnote{P. Amore and F. M. Fernandez, J. Phys. B 45, 235004 (2012).}