Singular Bohr-Sommerfeld Rules

Abstract

Abstract:The object of this paper is to present a geometrical formulation of the singular Bohr-Sommerfeld rules given in [8], in a more general context, including characteristic manifolds with several saddle points. Several examples are detailed and numerical checkings of the results are provided.

Singular Bohr–Sommerfeld rules for 2D integrable systems

“…The proofs for formulae (10), (11) and (12) are given in Section 2.4, but the formula for ε 1 is apparent in the proof of Theorem 2.4 below, and the formula for ε 2 can be directly checked using the fact that the subprincipal symbol is preserved under conjugation by an elliptic Fourier integral operator at a critical point of the principal symbol. ✷…”

“…We shall use this fact to construct from the sheaf (L, Λ o ) a new sheaf (L, G) on G ⊂ W (recall that W is the symplectic orbifold of Section 1.5) that will encode whether (L, Λ o ) has a global section (see Theorem 2.7). Generalising the construction of [10], to each point p ∈ G we associate the free moduleL(p) generated by the germs of standard basis at p, which will be of rank 1, as follows.…”

“…-In [10] the factor e −i π 4 was absent in the definition of T j (19). Its introduction here will greatly simplify the treatment of Maslov indices (see also [5]).…”

“…However we give no detail about the semiclassical treatment since a separation of variables shows that our formula reduces to two problems of one degree of freedom, which can be solved as in [10].…”

“…In this paper, we describe extensions of results of [10] to completely integrable semiclassical systems with 2 degrees of freedom. If H 1 , H 2 are two commuting h-pseudodifferential operators on a 2-manifold X, we introduce the momentum map F = (H 1 , H 2 ) : T X → R 2 where H j (j = 1, 2) is the principal symbol of H j ; we assume that F is a proper map.…”