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Propriétés spectrales d’opérateurs différentiels asymptotiques autoadjoints. (Spectral properties of asymptotical selfadjoint differential operators). (French) Zbl 0586.47033

The paper aims at investigating the semi-classical behaviour of operators in quantum mechanics and of certain relevant spectral properties of ”asymptotical” selfadjoint differential operators with application to 3- dimensional Schrödinger operators. The material is based, however, essentially on three recent works [J. Leray, Lagrangian analysis and quantum mechanics (1981; Zbl 0483.35002); V. P. Maslov, Théorie des perturbations, et méthodes asymptotiques (1972; Zbl 0247.47010); V. P. Maslov and M. V. Fedoryuk, Semi-classical approximation in quantum mechanics (1981; Zbl 0458.58001) and (1976; Zbl 0449.58002)]. Further the utilized technique: introduction of the ”principal symbol” for a class of ”oscillatory functions” associated with certain ”prequantified” families of Lagrangian varieties in Leray’s sense, and the developing of a purely algebraical ”symbolic calculus” together with precising its action on the related asymptotic differential operators - all this was inspired, as the authors stress it themselves, by some papers of Y. Colin de Verdière [Invent. Math. 43, 15-52 (1977; Zbl 0449.53040)], J. J. Duistermaat [Commun. Pure Appl. Math. 27, 207-281 (1974; Zbl 0285.35010)], L. Hörmander [Acta Math. 128, 183-269 (1972; Zbl 0232.47055)] and V. P. Maslov. Remark still that the main results (theorems 6.3, 6.6) are easy consequences of a proposition due to Duistermaat and Maslov.
Reviewer: M.Mikolas

MSC:

47B40 Spectral operators, decomposable operators, well-bounded operators, etc.
47F05 General theory of partial differential operators
81Q10 Selfadjoint operator theory in quantum theory, including spectral analysis
47B25 Linear symmetric and selfadjoint operators (unbounded)
58J50 Spectral problems; spectral geometry; scattering theory on manifolds
53C80 Applications of global differential geometry to the sciences
47A40 Scattering theory of linear operators
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References:

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[8] Maslov, V.P. and Fedoryuk, M.V. 1981. ”Semi–classical approximation in quantum mechanics”. Reidel. · Zbl 0458.58001
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