A symbol calculus for a class of pseudodifferential operators on \(S^ n\) and band asymptotics (Q2266389)

The author develops a global-intrinsic symbolic calculus for the ring of all the pseudo differential operators in \({\mathbb{S}}^ n\) which commute with the Laplace-Beltrami operator \(\Delta\). This calculus is applied to the Schrödinger operator \(\Delta +q\), \(q\in C^{\infty}({\mathbb{S}}^ n)\), for which certain invariants, the so-called band asymptotics, are computed. Related results on these invariants were proved by \textit{V. Guillemin} [Adv. Math. 42, 248-282 (1981; Zbl 0478.58029)] and by the author in his Ph. D. thesis (M.I.T. 1982).