Guillemin, Victor; Paul, Thierry Some remarks about semiclassical trace invariants and quantum normal forms. (English) Zbl 1213.58021 Commun. Math. Phys. 294, No. 1, 1-19 (2010). Let \(H:L^2(X)\rightarrow H^2(X)\) be a self-adjoint first order elliptic pseudo-differential operator on a smooth manifold \(X\). The authors consider the Schrödinger operator on \({\mathbb R}^n\) given by \(H=-h^2\Delta+V\) where \(V\rightarrow\infty\) as \(x\rightarrow\infty\) or more generally a self-adjoint semiclassical elliptic pseudo-differential operator whose symbol is proper as a map from the cotangent bundle of \(X\) to \({\mathbb R}\). The authors explore the connection between semi-classical and quantum Birkhoff canonical forms for such an operator and give a “non-symbolic” operator theoretic derivation of the quantum Birkhoff canonical form. They provide an explicit recipe for expressing the quantum Birkhoff canonical forms in terms of the semi-classical Birkhoff canonical form and develop a purely quantum mechanical approach to the theory of Birkhoff canonical forms in which symbolic expressions get replaced by operator theoretic expansions and estimates involving Hermite functions – this is a local version of the Rayleigh-Schrödinger perturbation formalism. Reviewer: Peter B. Gilkey (Eugene) Cited in 8 Documents MSC: 58J50 Spectral problems; spectral geometry; scattering theory on manifolds 58J40 Pseudodifferential and Fourier integral operators on manifolds 81Q20 Semiclassical techniques, including WKB and Maslov methods applied to problems in quantum theory Keywords:Birkhoff canonical form; pseudo-differential operator; classical dynamical system; Schrödinger operator; microlocal; Zelditch-type algorithm; Rayleigh-Schrödinger perturbation; Hamiltonian; Fourier integral operator; total Weyl symbol PDFBibTeX XMLCite \textit{V. Guillemin} and \textit{T. Paul}, Commun. Math. Phys. 294, No. 1, 1--19 (2010; Zbl 1213.58021) Full Text: DOI arXiv References: [1] Bridges, T.J., Cushman, R.H., MacKay, R.S.: Dynamics near an irrational collision of eigenvalues for symplectic mappings. Lamgford, W.F. (ed.) In: Normal Forms and Homoclinic Chaos. Fields Inst. Commun. 4, Providence, RI: Amer. Math. Soc., 1995, pp. 61–79 · Zbl 0833.58012 [2] Chazarain J.: Formule de Poisson pour les variétés Riemanniennes. Invent. Math. 24, 65–82 (1974) · Zbl 0281.35028 · doi:10.1007/BF01418788 [3] Colin de Verdière Y.: Spectre du Laplacien et longueurs des géodésiques périodiques. Compos. Math. 27, 83–106 (1973) · Zbl 0272.53034 [4] Duistermaat J.J., Guillemin V.: The spectrum of positive elliptic operators and periodic bicharacteristics. Inv. Math. 29, 39–79 (1975) · Zbl 0307.35071 · doi:10.1007/BF01405172 [5] Guillemin V.: Wave-trace invariants. Duke Math. J. 83, 287–352 (1976) · Zbl 0858.58051 · doi:10.1215/S0012-7094-96-08311-8 [6] Gutzwiller M.: Periodic orbits and classical quantization conditions. J. Math. Phys. 12, 343–358 (1971) · doi:10.1063/1.1665596 [7] Iantchenko A., Sjöstrand J., Zworski M.: Birkhoff normal forms in semi-classical inverse problems. Math. Res. Lett. 9, 337–362 (2002) · Zbl 1258.35208 [8] Paul, T., Uribe, A.: Sur la formule semi-classique des traces. C.R. Acad. Sci Paris 313, I, 217–222 (1991) · Zbl 0738.58046 [9] Paul T., Uribe A.: The semi-classical trace formula and propagation of wave packets. J. Funct. Anal. 132, 192–249 (1995) · Zbl 0837.35106 · doi:10.1006/jfan.1995.1105 [10] Robert D.: Autour de l’approximation semi-classique. Basel-Boston, Birkhäuser (1987) · Zbl 0621.35001 [11] Zelditch S.: Wave invariants at elliptic closed geodesics. Geom. Funct. Anal. 7, 145–213 (1997) · Zbl 0876.58010 · doi:10.1007/PL00001615 [12] Zelditch S.: Wave invariants for non-degenerate closed geodesics. Geom. Funct. Anal. 8, 179–217 (1998) · Zbl 0908.58022 · doi:10.1007/s000390050052 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.