These notes are based on six Fermi Lectures held at the Scuola
Normale Superiore in Pisa in March and April 1981. The topics
treated depend on basic concepts of classical mechanics, elementary
geometry, complex analysis as well as spectral theory and are meant
for mathematicians and theoretical physicists alike. These lectures
weave together a number of threads from various fields of
mathematics impinging on the subject of inverse spectral theory. I
did not try to give an overview over this fast moving subject but
rather tie various aspects together by one guiding theme: the
construction of all potentials for the one-dimensional Schrodinger
equation which gives rise to finite band potentials, which is done
by reducing it to solving a system of differential equations. In
fact, we will see that the problem of finding all almost periodic
potentials having finitely many intervals as its spectrum is
equivalent to the study of the geodesics on an ellipsoid. To make
this connection clear we have carried together several facts from
classical mechanics and from spectral theory and we give a
self-contained exposition of the construction of these finite band
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