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The vector field20 Θ(H|Q) = det XH = −2Hxx (Q) has integral curves d2 ln Θ(H|Q). dx2 Ordinary infinitesimal addition at parameter λ is −2Gxx (λ |Q); this corresponds to H(λ) = 1/(λ − λ ). 3. 21 When is an additive class a finite-dimensional manifold? How do you translate this geometric property in the space of all operators into analytic properties of individual operators? Remarkably, this question has a complete answer. 19The superscript zero is dropped for convenience. 20H is the kernel of the integral operator H(Q).

And spectral theory in a masterful way, Henry APPENDIX: HAMILTONIAN MECHANICS 29 succeeds in proving that Q is a Hill opera- ideas, even if imperfectly and slowly. Thank tor, which means: it has a period 1 potential you, Henry, for making mathematics so much whose divisor {pj } is distributed one per gap. fun. 5. Other Operators. Modifications and combinations of inverse scattering and the band-gap spectral theory have been worked out for potentials with rapid decay to nonzero constants at |x| = ∞ and for rapidly decaying perturbations of finite-gap potentials, but not, I believe, for local perturbations of general Hill operators.

Jacobi Inversion. Hill oper- tential q can be found from the so-called trace ators are a chapter in the theory of the formula, (λ − λm ), the multivalued function 2g g product being finite or infinite. 4) λm − 2 μj (x), q(x) = direct introduction to this circle of ideas m=0 j=1 starts with condition (a). 1), provided we are able μj of M Φ = 4λΦ . The coefficients can be deter- to express the symmetric function the upper limits in terms of the right side. mined recursively, down to the constant term, Riemann’s theory of his theta function was which does not vanish automatically.