Perturbative expansions in quantum mechanics

Garay, Mauricio D.

Annales de l'Institut Fourier, Tome 59 (2009) no. 5, pp. 2061-2101.

Nous démontrons un théorème de déformation verselle analytique pour l’algèbre de Heisenberg dans le cas $D=1$. Nous définissons le spectre d’un élément dans cette algèbre. La quantification du lemme de Morse montre que les séries perturbatives du spectre de l’oscillateur harmonique deviennent analytique après une transformation de Borel formelle.

We prove a $D=1$ analytic versal deformation theorem in the Heisenberg algebra. We define the spectrum of an element in the Heisenberg algebra. The quantised version of the Morse lemma already shows that the perturbation series arising in a perturbed harmonic oscillator become analytic after a formal Borel transform.

Classification : 81Q15
Keywords: Harmonic oscillator, Borel summability, micro-local analysis, non-commutative geometry
Mot clés : oscillateur harmonique, sommabilité de Borel, analyse semi-classique, formes normales

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