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# Dynamics of Partial Differential Equations

## Volume 2 (2005)

### Number 2

### The inverse periodic spectral theory of the Euler-Bernoulli equation

Pages: 127 – 148

DOI: http://dx.doi.org/10.4310/DPDE.2005.v2.n2.a2

#### Author

#### Abstract

The Floquet (direct spectral) theory of the periodic Euler-Bernoulli equation has been developed by the author in [37], [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).

#### Keywords

Euler-Bernoulli (or beam) operator, Euler-Bernoulli equation for the vibrating beam, Hill’s operator, periodic coefficients, Floquet theory, spectrum, pseudospectrum, multipoint eigenvalue problem, inverse periodic spectral theory, Abel’s theorem

#### 2010 Mathematics Subject Classification

Primary 34A55, 34B05. Secondary 34B10, 34B30, 34L40, 74B05.