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Some remarks about semiclassical trace invariants and quantum normal forms. (English) Zbl 1213.58021

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.

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
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