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Bohr-Sommerfeld rules to all orders. (English) Zbl 1080.81029
Summary: The goal of this paper is to give a rather simple algorithm which computes the Bohr-Sommerfeld quantization rules to all orders in the semi-classical parameter $$h$$ for a semi-classical Hamiltonian $$\widehat{H}$$ on the real line. The formula gives the high-order terms in the expansion in powers of $$h$$ of the semi-classical action using only integrals on the energy curves of quantities which are locally computable from the Weyl symbol. The recipe uses only the knowledge of the Moyal formula expressing the star product of Weyl symbols. It is important to note that our method assumes already the existence of Bohr-Sommerfeld rules to any order (which is usually shown using some precise Ansatz for the eigenfunctions, like the WKB-Maslov Ansatz) and the problem we address here is only about ways to compute these corrections. Existence of corrections to any order to Bohr-Sommerfeld rules is well known and can be found for example in [B. Helffer and D. Robert, Ann. Inst. Henri Poincaré, Phys. Théor. 41, 291–331 (1984; Zbl 0565.35082)] and [San Vũ Ngoc, Sur le spectre des systèmes complètement intégrales semi-classiques avec singularités (French), Doctor-thesis, Grenoble (1998), Section 4.5].
Our way to get these high-order corrections is inspired by A. Voros’s thesis [Asymptotic $$\hbar$$-expansions of stationary quantum states, Doctor-thesis, Orsay (1977); Ann. Inst. H. Poincaré, Nouv. Sér., Sect. A 26, 343–403 (1977)]. The reference [P. N. Agyres, Physics 2, 131–199 (1965)], where a very similar method is sketched, was given to us by A. Voros. We use also in an essential way the nice formula of Helffer-Sjöstrand expressing $$f(\widehat {H})$$ in terms of the resolvent.

##### MSC:
 81S10 Geometry and quantization, symplectic methods 81Q20 Semiclassical techniques, including WKB and Maslov methods applied to problems in quantum theory
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