Supersymmetric generalized power functions

Abstract

Complex-valued functions defined on a finite interval [a, b] generalizing power functions of the type (x−x0)n for n ≥ 0 are studied. These functions called Φ-generalized powers, Φ being a given nonzero complex-valued function on the interval, were considered to construct a general solution representation of the Sturm–Liouville equation in terms of the spectral parameter [V. V. Kravchenko and R. M. Porter, Math. Methods Appl. Sci. 33(4), 459–468 (2010)]. The Φ-generalized powers can be considered as natural basis functions for the one-dimensional supersymmetric quantum mechanics systems taking Φ=ψ02, where the function ψ0(x) is the ground state wave function of one of the supersymmetric scalar Hamiltonians. Several properties are obtained such as Φ-symmetric conjugate and antisymmetry of the Φ-generalized powers, a supersymmetric binomial identity for these functions, a supersymmetric Pythagorean elliptic (hyperbolic) identity involving four Φ-trigonometric (Φ-hyperbolic) functions, as well as a supersymmetric Taylor series expressed in terms of the Φ-derivatives. We show that the first n Φ-generalized powers are a fundamental set of solutions associated with nonconstant coefficient homogeneous linear ordinary differential equations of order n + 1. Finally, we present a general solution representation of the stationary Schrödinger equation in terms of geometric series where the Volterra compositions of the first type are considered.