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Zbl 1207.81026
Garay, Mauricio D.
Perturbative expansions in quantum mechanics. (English)
[J] Ann. Inst. Fourier 59, No. 5, 2061-2101 (2009). ISSN 0373-0956; ISSN 1777-5310/e

Denote by $\widehat Q$ the non-commutative algebra over $\mathbb{C}$ consisting of formal power series in the variables $p,q,\hbar$ satisfying the commutation relations $[p,q]=\hbar$, $[\hbar,p]=[\hbar,q]=0$; its elements have a representation as differential operators acting on the module~$\mathbb{C}[[\hbar,z]]$ via $p:=\hbar \partial_z$ and $q=z$. In this paper, the author studies convergence of the perturbation series in the auxiliary parameter~$t$ for the spectrum of the operators of the form~$H_0+tH_1$ with $H_0\in \hat Q$ and $H_1\in \hat Q[[t]]$. The motivation stems from the paper by [{\it W.~Heisenberg}, ``Über quantentheoretische Umdeutung kinematischer und mechanischer Beziehungen", Z. f. Physik 33, 879--893 (1925; JFM 51.0728.07)], who studied such an expansion for the anharmonic oscillator $H(p,q)=(p^2+q^2)/2 + tq^4/4$, and later developments by Born and Jordan. The author discusses analogous problem in the abstract setting of the analytic category. He establishes the quantum Morse lemma and finiteness of the deformation module and then proves the quantum versal deformation theorem, which allows him to justify the conjecture by {\it Y. Colin de Verdière} [Duke Math. J. 116, No. 2, 263--298 (2003; Zbl 1074.53066)]. By applying his abstract results to the anharmonic oscillator $H(p,q)$ the author proves that the perturbation series becomes analytic after the Borel transform.
[Rostyslav Hryniv (Lviv)]

MSC 2000:: *81Q15 Perturbation theories for operators and differential equations
32S30 Deformations of singularities (analytic spaces)

Keywords: harmonic oscillator; Borel summability; micro-local analysis; non-commutative geometry

Citations: Zbl 1074.53066; JFM 51.0728.07

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