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The eigenvalues’ function of the family of Sturm-Liouville operators and the inverse problems

We study the direct and inverse problems for the family of Sturm-Liouville operators, generated by fixed potential q and the family of separated boundary conditions. We prove that the union of the spectra of all these operators can be represented as a smooth surface (as the values of a real analytic function of two variables), which has specific properties. We call this function ”the eigenvalues function of the family of Sturm-Liouville operators (EVF)”. From the properties of this function we select those, which are sufficient for a function of two variables be the EVF a family of Sturm-Liouville operators.