A characterization of the spectrum of Hill's operator

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.
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