Rayleigh-Schroedinger perturbation theory generalized to eigen-operators in non-commutative rings

Abstract : A perturbation scheme to find approximate solutions of a generalized spectral problem is presented. The spectral problem is generalized in the sense that the ``eigenvalues'' searched for, are not real numbers but operators in a non-commutative ring, and the associated ``eigenfunctions'' do not belong to an Hilbert space but are elements of a module on the non-commutative ring. The method is relevant wherever two sets of degrees of freedom can be distinguished in a quantum system. This is the case for example in rotation-vibration molecular spectroscopy. The article clarifies the relationship between the exact solutions of rotation-vibration molecular Hamiltonians and the solutions of the effective rotational Hamiltonians derived in previous works. It also proposes a less restrictive form for the effective dipole moment than the form considered by spectroscopists so far.
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Patrick Cassam-Chenaï. Rayleigh-Schroedinger perturbation theory generalized to eigen-operators in non-commutative rings. Journal of Mathematical Chemistry, Springer Verlag (Germany), 2011, 49 (4), p. 821-835. ⟨10.1007/s10910-010-9779-y⟩. ⟨hal-00506483⟩

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