Given a set of $N$ increasing real numbers $\{E_1, E_n, \cdots, E_N \}$, is it always possible to find a potential $V(x)$ such that the set of $\{E_j\}$ are the lowest eigenvalues of the corresponding one-dimensional Schrödinger equation for given mass?
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1Possible duplicates: https://physics.stackexchange.com/q/13480/2451 and links therein. – Qmechanic Aug 29 '19 at 16:37
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Yes, you may always craft a reflection less potential through SUSYQM starting from the shallowest level. A quick review you might appreciate is Kwong & Rosner 1986. – Cosmas Zachos Aug 29 '19 at 16:40
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thanks! Both the reviews and the previous answer are very relevant. I wonder, though, how 'physical' some of the isospectral potential are. For example, some may have a lot of narrow peaks in order to shift a particular level. I wonder what would happen if we imposed limitations on the derivatives of the potentials to prevent 'peaks' or other near singularities. – Lorents Sep 02 '19 at 14:10