ON THE EQUIVALENCE OF THE INTEGRAL AND THE DIFFERENTIAL EXACT SOLUTION GENERATION METHODS FOR THE ONE-DIMENSIONAL SCHRODINGER-EQUATION

It is demonstrated that the integral exact solution generation methods for the one-dimensional Schrodinger equation based on the Gelfand-Lev itan formalism are in some cases equivalent to the differential ones b ased on the n-order Darboux transformation. Some new exact solvable po tentials are generated from the effective Coulomb potential and the ha rmonic oscillator potential. A new form of n-soliton potential (i.e. r eflectionless potential with n discrete energy levels disposed in a de sirable manner) based on an explicit expression for an n-order Wronski determinant constructed from hyperbolic functions and its orthonormal discrete spectrum eigenfunctions are given.