A fully algebraic approach to constructing one-dimensional reflectionless potentials with any number (N) of bound states is described. A simple and easily applicable general formula is derived, using the methods of the theory of determinants. In particular, useful properties of special determinants – the alternants – have been exploited. The modified determinant that uniquely fixes the potential contains only 2N−1 terms, which is a huge win compared to the N! terms of the original expansion. Moreover, the modified determinant can be very easily evaluated using the properties of alternants. To this end, two useful theorems have been proved. The main formula takes an especially simple form if one aims to reconstruct a symmetric reflectionless potential. Several examples are presented to illustrate the efficiency of the method.
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