Results in Physics (Mar 2024)
Supersymmetric quantum mechanics of hypergeometric-like differential operators
Abstract
Systematic iterative algorithms, that are solely dictated by the principles of supersymmetric quantum mechanics and do not rest on any input from the traditional methods, are developed for constructing the discrete eigen-spectra of a generic principal hypergeometric-like differential operator and also for iteratively building its associated hypergeometric-like differential operator as well as the eigen-spectra of the latter in one go. These pure algebraic algorithms reveal the simple supersymmetric quantum mechanical reason for why, for the same principal hypergeometric-like differential operator, there exist simultaneously a tower of principal as well as of associated special functions in their canonical forms. That is, two distinct types of quantum momentum operators as well as the initial superpotentials are rooted in this type of operator, i.e. the same hypergeometric-like operator admits two distinct supersymmetric factorizations, and each of these superpotentials/supersymmetric factorizations can proliferate into a hierarchy of descendant superpotentials/supersymmetric factorizations in the same shape-invariant fashion. These algebraic algorithms can be implemented in equal efficiency either directly in the non-standard x-coordinate representation of the underlying supersymmetric quantum mechanics of the hypergeometric-like differential operator or in its standard y/z-coordinate representation, and the two representations are isomorphically connected to each other through two types of active supersymmetrization transformations plus the accompanying momentum operator maps, which are readily constructed by supersymmetrizing the two types of trivial asymmetric factorizations of that principal hypergeometric-like operator in an elementary algebraic way. Due to their conceptual simplicity, relatively high efficiency and nature of doing analysis without analysis, these algorithms make the hypergeometric-like special functions and their associated cousins not so special at all as in their guises out of the popular traditional methods, and therefore could become the hopefuls for supplanting the latter.
