On the convergence of reflectionless approximations to confining potentials |
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Authors: | Jonathan F Schonfeld Waikwok Kwong Jonathan L Rosner C Quigg H.B Thacker |
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Affiliation: | Wolfson College, Oxford, OX2 6UD England |
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Abstract: | The construction of reflectionless potentials supporting a prescribed spectrum of Schrödinger bound states is discussed and related to the inverse problem for confining potentials. A simple formula is derived for the Jost solution in a one-dimensional reflection-less potential with N bound states. This leads to compact expressions for the potential and the bound-state wavefunctions in terms of the bound-state energies. For symmetric potentials, N-fold product formulas are obtained for bound-state wavefunctions and their slopes at the origin. Corresponding quantities in a confining potential are given by infinite products. Comparison of the finite-product and infinite-product expressions allows a demonstration of the convergence of the reflectionless results to the confining potential results as N → ∞. Several sum rules satisfied by the reflectionless potential at the origin are applied to numerical studies of convergence. |
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