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A symbol calculus for a class of pseudodifferential operators on \(S^ n\) and band asymptotics. (English) Zbl 0561.35082

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 V. Guillemin [Adv. Math. 42, 248-282 (1981; Zbl 0478.58029)] and by the author in his Ph. D. thesis (M.I.T. 1982).
Reviewer: L.Rodino

MSC:

35S05 Pseudodifferential operators as generalizations of partial differential operators
35J10 Schrödinger operator, Schrödinger equation
47Gxx Integral, integro-differential, and pseudodifferential operators

Citations:

Zbl 0478.58029
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References:

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