On the band gap structure of Hill's equation |
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Authors: | G. Allaire R. Orive |
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Affiliation: | a Centre de Mathématiques Appliquées, Ecole Polytechnique, 91128 Palaiseau Cedex, France b Instituto de Matemáticas y Física Fundamental, Consejo Superior de Investigaciones Científicas, Serrano 123, 28006 Madrid, Spain |
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Abstract: | We revisit the old problem of finding the stability and instability intervals of a second-order elliptic equation on the real line with periodic coefficients (Hill's equation). It is well known that the stability intervals correspond to the spectrum of the Bloch family of operators defined on a single period. Here we propose a characterization of the instability intervals. We introduce a new family of non-self-adjoint operators, formally equivalent to the Bloch ones but with an imaginary Bloch parameter, that we call exponential. We prove that they admit a countable infinite number of eigenvalues which, when they are real, completely characterize the intervals of instability of Hill's equation. |
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