Supersymmetric features of the Error and Dawson's functions

TL;DR

This paper explores supersymmetric quantum mechanics methods to derive analytical approximations for the error and Dawson's functions, revealing new mathematical relationships and solutions.

Contribution

It introduces novel SUSY-based approaches to approximate and analyze the error and Dawson's functions through eigenvalue problems.

Findings

Analytical approximation of the error function using hyperbolic tangent series

Identification of SUSY relations between Dawson's differential equation and harmonic oscillator eigenvalue problems

New insights into the mathematical structure of special functions via SUSY methods

Abstract

Following a letter by Bassett, we show first that it is possible to find an analytical approximation to the error function in terms of a finite series of hyperbolic tangents from the supersymmetric (SUSY) solution of the Poschl-Teller eigenvalue problem in quantum mechanics (QM). Afterwards, we show that the second order differential equation for the derivatives of Dawson's function can be found in another SUSY related eigenvalue problem, where the factorization of the simple harmonic oscillator Hamiltonian renders the wrong-sign Hermite differential equation, and that Dawson's second order differential equation possess a singular SUSY type relation to this equation.