Suppose one is given with a sequence $S$ of non-negative real numbers $0\leq\lambda_0\leq \lambda_1\leq\dots\leq \lambda_n\leq\dots$. Under what conditions on $S$, is it possible to construct a Linear differential operator $D$ on the space of square integrable functions on the real line, such that the eigenvalue spectrum of $D$ is given by $S$ ? Is there any way to explicitly construct such an operator ?



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