Trace Formulas and the Canonical 1-Form

Abstract

This paper studies the canonical 1-form of symplectic geometry in the context of the (defocussing) cubic Schrodinger system. The phase space is populated by pairs QP of smooth functions of period 1, equipped with the classical 1-form QdP = ∫ ¹₀ [Q(x)dP(x)]dx. The introduction of canon- ically paired coordinates Q _n P _n : n ∈ ℤ, as in Sections 2 and 6 below, suggests the identity QdP = Σ_ℤ Q _n dP _n , up to an additive exact form, and this may be verified, as in Sections 5 and 6, with the help of new trace formulas, derived in what I believe to be a new way; see, especially Section 4, nos. 4 and 5. The discussion could be carried over to sine/sh-Gordon, etc.; compare Section 7 where this is done for KdV.

This work was performed at the Courant Institute of Mathematical Sciences with the partial support of the National science Foundation, under NSF Grant No. DMS- 9112664, which is gratefully acknowledged.