Candelpergher, B.; Nosmas, J. C. Propriétés spectrales d’opérateurs différentiels asymptotiques autoadjoints. (Spectral properties of asymptotical selfadjoint differential operators). (French) Zbl 0586.47033 Commun. Partial Differ. Equations 9, 137-167 (1984). 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 Cited in 4 Documents 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 Keywords:spectral properties of asymptotical selfadjoint differential operators; semi-classical behaviour of operators in quantum mechanics; 3-dimensional Schrödinger operators; principal symbol; oscillatory functions; symbolic calculus Citations:Zbl 0483.35002; Zbl 0247.47010; Zbl 0458.58001; Zbl 0449.58002; Zbl 0449.53040; Zbl 0285.35010; Zbl 0232.47055 PDFBibTeX XMLCite \textit{B. Candelpergher} and \textit{J. C. Nosmas}, Commun. Partial Differ. Equations 9, 137--167 (1984; Zbl 0586.47033) Full Text: DOI References: [1] Abraham, R. and Marsden, J.E. 1978. ”Foundations of Mechanics”. Benjamin. [2] Duistermaat J.J., Courant Inst (1973) [3] Eckmann, J.P. and Seneor, R. 1976. ”Comm. inMaths Physics”. [4] Guillemin V., Maths Surveys 16 (1977) [5] Helffer, B. and Robert, D. 09 1977. ”comm. au Colloque de Sofia”. 09, [6] Leray J., Collége de France (1976) [7] Maslov, V.P. 1972. ”Méthodes asymptotiques et théorie des perturbations”. Dundo. [8] Maslov, V.P. and Fedoryuk, M.V. 1981. ”Semi–classical approximation in quantum mechanics”. Reidel. · Zbl 0458.58001 [9] Nosmas, J.C. 10 1982. ”Note au C.R.A.S”. Vol. 295, 10, [10] Voros A., Orsay (1977) [11] Candelpergher, B. and Nosmas, J.C. 1980. ”Note au C.R.A.S”. Vol. 291, This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.