Supersymmetric generalized power functions

Abstract

Complex-valued functions defined on a finite interval $[a,b]$ generalizing power functions of the type $(x-x_0{)}^n$ for $n\ge 0$ are studied. These functions called $\hat{I}¦$-generalized powers, $\hat{I}¦$ being a given nonzero complex-valued function on the interval, were considered to contruct a general solution representation of the Sturm-Liouville equation in terms of the spectral parameter \cite{kravchenko2008, kravporter2010}. The $\hat{I}¦$-generalized powers can be considered as a natural basis functions for the one-dimensional supersymmetric quantum mechanics systems taking $\hat{I}¦=\ddot{I}{ˆ}_0^2$, where the function $\ddot{I}{ˆ}_0(x)$ is the ground state wave function of one of the supersymmetric scalar Hamiltonians. Several properties are obtained such as $\hat{I}¦$-symmetric conjugate and antisymmetry of the $\hat{I}¦$-generalized powers, a supersymmetric binomial identity for these functions, a supersymmetric Pythagorean elliptic (hyperbolic) identity involving four $\hat{I}¦$-trigonometric ($\hat{I}¦$-hyperbolic) functions as well as a supersymmetric Taylor series expressed in terms of the $\hat{I}¦$-derivatives. We show that the first $n$ $\hat{I}¦$-generalized powers are a fundamental set of solutions associated with a nonconstant coefficients 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 is considered.

29 pages, 4 figures