By Jürgen Moser

Those notes are in response to six Fermi Lectures held on the Scuola Normale Superiore in Pisa in March and April 1981. the themes handled depend upon simple strategies of classical mechanics, common geometry, complicated research in addition to spectral thought and are intended for mathematicians and theoretical physicists alike. those lectures weave jointly a few threads from a variety of fields of arithmetic impinging almost about inverse spectral conception. i didn't try and supply an outline over this fast paced topic yet particularly tie a number of elements jointly by way of one guiding subject: the development of all potentials for the one-dimensional Schrödinger equation which provides upward thrust to finite band potentials, that's performed by means of decreasing it to fixing a process of differential equations. in reality, we'll see that the matter of discovering all virtually periodic potentials having finitely many periods as its spectrum is resembling the research of the geodesics on an ellipsoid. To make this connection transparent we now have carried jointly a number of proof from classical mechanics and from spectral idea and we provide a self-contained exposition of the development of those finite band potentials.

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K-1 Fk z-a,, . 1, A. 2, ... lso. re distinct. The leaves of the foliation F,. lso be defined via. ;(ro, y) = {11 (j = 1, ... , ... , n -1) GEODESICS ON AN ELLIPSOID ETC. 27 if {11, ... , fln-1 are the zeros of in other words, if the {11, {121 ... ck; '1>p, = ... _, = O} . dries 12p, (j = 1, 2, ... , n -1). -1= 0. 1, A. 2, ... - 2 as n -1 commuting integrals on the tangent bundle of the ellipsoid. e. to unit velocity. 1, A. 2, ... 10) the lines which are common tangents of 12p,, ... - 1 = 0 initially remain common tangent for all time.

0. s the «rotation number ». tion of a(A) for real ).. 2. Let ). 2). ) ; 11! in parti

Dense denumerable set, instead of the integers. 6 oc(A) is a monotone increasing function on the real axis and therefore doc can be viewed as a measure. 8. The spectritm a(L) agrees with the suppot•t of the meas- ure dac(A). This generalizes the characterization of the bands in the periodic case. 5. - The Green's function and a trace formula. 8) R;i. 2) which are in L 2 (0, oo), L 2 ( - oo, 0) respectively. It is known since Hermann Weyl [30] that 'I'+' 'I'- exist and are up to a factor uniquely determined.

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