The inverse periodic spectral theory of the Euler-Bernoulli equation

The Floquet (direct spectral) theory of the periodic Euler-Bernoulli equation has been developed by the author in [371, [41], and [38]. A particular case of the inverse problem has been studied in [39]. Here we focus on the inverse periodic spectral problem. A key ingredient is an extended version of Abel's theorem for the existense of meromorphic functions on Riemann Surfaces. To avoid technicalities, we have assumed that the Floquet multiplier has finitely many branch points (in the Hill operator case this corresponds to the assumption that the spectrum has finitely many gaps).