Those notes are in keeping with six Fermi Lectures held on the Scuola Normale Superiore in Pisa in March and April 1981. the themes taken care of rely on uncomplicated techniques of classical mechanics, undemanding geometry, complicated research in addition to spectral conception and are intended for mathematicians and theoretical physicists alike. those lectures weave jointly a few threads from quite a few fields of arithmetic impinging with reference to inverse spectral thought. i didn't attempt to provide an outline over this fast-paced topic yet quite tie numerous elements jointly via one guiding topic: the development of all potentials for the one-dimensional Schrödinger equation which provides upward push to finite band potentials, that is performed by means of lowering it to fixing a procedure of differential equations. in reality, we'll see that the matter of discovering all nearly periodic potentials having finitely many periods as its spectrum is similar to the research of the geodesics on an ellipsoid. To make this connection transparent we've got carried jointly numerous evidence from classical mechanics and from spectral idea and we supply a self-contained exposition of the development of those finite band potentials.
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