Abstract
The Floquet (direct spectral) theory of the periodic Euler-Bernoulli equation has been developed by the author in [19], [21], and [20]. Here we begin a systematic study of the inverse periodic spectral theory, in the spirit of the corresponding theory of the second-order operator, namely the Hill’s operator. Our main result is that, if there are no pseudogaps (equivalently, if the BlochFloquet variety is reducible in a certain sense), then the Euler-Bernoulli operator is the square of a second-order (Hill-type) operator. This result had been conjectured by the author, in his earlier works.
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