Equivariant Asymptotics for Bohr-Sommerfeld Lagrangian Submanifolds

A note on scaling asymptotics for Bohr-Sommerfeld Lagrangian submanifolds

“…This agrees with Corollary 1.1 of [5] to leading order but gives a clearer picture of the asymptotic expansion, as well as an explicit estimate on the remainder.…”

“…The purpose of this paper is to improve an expansion in [5] for the asymptotics associated to Bohr-Sommerfeld Lagrangian submanifolds of a compact Hodge manifold, in the context of geometric quantization (see e.g. [1], [2], [6], [9]).…”

“…Referring to §2 of [5] for a more complete description of the prelimiaries involved, we recall that if Λ ⊆ X is a compact Legendrian submanifold and λ is a half-density on it, there is a naturally induced generalized half-density δ Λ,λ on X supported on Λ. Following [2], we can then define a sequence of CR functions…”

“…On the other hand, the asymptotic concentration of the u k 's along S 1 · Λ poses an interesting problem, already considered in Theorem 3.12 of [2]. This theme was revisited in [5] in a somewhat different technical setting; in particular, Corollary 1.1 of [5] shows that the scaling asymptotics of u k (to be defined shortly) near any x ∈ S 1 · Λ admit an asymptotic expansion, and explicitly computes the leading order term. We shall presently give a more precise description of this expansion as a function on the tangent space of M at m = π(x).…”

“…Namely, we shall show that this asymptotic expansion may be factored as an exponentially decaying term in the component w ⊥ of w ∈ T m M orthogonal to Λ times an asymptotic expansion with polynomial coefficients in w (more precisely, the expansion is generally given by a finite sum of terms of this form, one from each branch of Λ projecting to m); the exponential term also contains a symplectic pairing between w ⊥ and the component of w along Λ, w . Furthermore, we shall give some relevant remainder estimates not mentioned in [5].…”