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By Jürgen Moser
Those notes are according to six Fermi Lectures held on the Scuola Normale Superiore in Pisa in March and April 1981. the themes handled rely on easy innovations of classical mechanics, easy geometry, advanced research in addition to spectral thought and are intended for mathematicians and theoretical physicists alike. those lectures weave jointly a couple of threads from a number of fields of arithmetic impinging near to inverse spectral concept. i didn't attempt to supply an summary over this fast paced topic yet fairly tie numerous facets jointly by way of one guiding subject matter: the development of all potentials for the one-dimensional Schrödinger equation which supplies upward thrust to finite band potentials, that's performed by means of lowering it to fixing a method 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 corresponding to the research of the geodesics on an ellipsoid. To make this connection transparent we now have 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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Extra info for Integrable Hamiltonian systems and spectral theory
Example text
Agc, Tcubner, Berlin, Leipzig (1910), p. 658. A is a symmetric matrix with distinct eigenvalues. Aq, q) - lql 2 • This also is an integrable system which is seen as follows: We extend also this system to a system in R 2". 13) to the tangent bundle of sn-l and it suffices to show that this system is integrable. 1 again: Expanding the rational function . /) + jxj2jyj2- (x, y)B} + ... (p, q) = ~ + 2H(~, p) z z + .... 14) H " = ! ; akFk(p, q) . 9) with (x, y) replaced 29 GEODESICS ON AN ELLIPSOID ETC.
Moreover, all these integrals commute. 10. 1; q), w(A. 2 ; q)} = 0 The calculation can be found in [12]. ; q) to the KdV equation we study its asymptotic behavior for A. -+ - oo. g. G1 = q/2. ) on the diagonal always satisfies. re polynomials in q and its derivatives. re called «local functionals ». As a. 10 we see that An explicit calculation yields Wa 1M(l2q" + qa) • = 16 9 52 SECTION 4 So 16w3 = H corresponds to the KdV equation. The w 1 represent the infinitely many conservation laws of this equation.
Here we will study the simplest almost periodic potentials for which the spectrum is very simple and consists of finitely many intervals. We turn to the inverse problem to determine all potentials having such finite band structure. 11) of Neumann. 1) we refer to these N + 1 intervals as «bands » and the complementary N + 1 intervals as «gaps ». We reformulate the problem more precisely. For any real potential q SECTION 5 54 the Green's function is real on the part of the real axis which belongs to the resolvent set.