Geodesic
A~characterization of the spectrum of Hill's operator
Sbornik. Mathematics, Tome 26 (1975) no. 4, pp. 493-554

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This article contains a complete derivation of necessary and sufficient conditions which a given sequence of intervals must satisfy in order that a Hill differential operator $L[y]=-y''+v(x)y$, with real, periodic potential $v(x)$, exist, whose spectrum coincides with this sequence of intervals. The proof is based on a specific representation of entire functions $u(z)$ such that the equation $u^2(z)=1$ has only real roots, conformal mappings having properties associated with this representation, and refined asymptotic formulas for the eigenvalues of certain boundary value problems. Figures: 4. Bibliography: 17 titles. 
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     author = {V. A. Marchenko and I. V. Ostrovskii},
     title = {A~characterization of the spectrum of {Hill's} operator},
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     year = {1975},
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V. A. Marchenko; I. V. Ostrovskii. A~characterization of the spectrum of Hill's operator. Sbornik. Mathematics, Tome 26 (1975) no. 4, pp. 493-554. http://geodesic.mathdoc.fr/item/SM_1975_26_4_a4/