Perturbation theory of quantum resonances
Ivana Paidarova2016, Theoretical Chemistry Accounts
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Abstract
The partitioning theory provides optical potentials which lead to the resonance energies. The state of the theory in the eighties can be found in the book "Theory of resonances" of Kukulin et al. [3]. The basic principles presented in this book still hold: use of projectors and analytical continuation. Since that time, the theoretical developments have been influenced by the fast development of computers and the theory entered into the quickly expanding field of non-Hermitian quantum mechanics [4]. Jolicard and Austin [5] incorporated optical potentials in computational schemes, and the justification of their method was given by Riss and Meyer [6]. To avoid confusion, we do not use the term optical potential here, but adopt the more suitable expression complex absorbing potential (CAP), as proposed in Ref. [6]. The acronym CAP refers to an energy-independent complex potential added to the Hamiltonian (see review article [7]). For the past two decades, we contributed to the development of approaches combining CAP and perturbation theory [8-12]. In this article, we recognize a resonance as a pole of a Green's function. In Sect. 2, we first recall the definition of the wave function and the energy in the framework of the partitioning technique. Then, a complex absorbing potential written in the form V CAP = −ı ǫ is added to the Hamiltonian. The operator ǫ generalizes the quantity ǫ of collision theory. The parameter in V CAP anticipates that V CAP does not only produce analytic continuation, but that it is also a perturbation operator. The self-energy is expanded in powers of , and the equivalence between this expansion and a Taylor series is demonstrated in Sect. 3 which focuses on our new findings. Finally, the introduction of a convergence operator allows to discuss the convergence properties of these series. Section 4 is devoted to numerical illustrations. First, a discretized N-dimensional Fano model is used to check the accuracy and the convergence properties of the energies. It is shown Abstract We propose a contribution to the theory of quantum resonances that combines complex absorbing potentials (CAP) with standard perturbation theory. We start from resolvents that depend on two variables, the complex energy z and a perturbation parameter. The wave functions and the energies of the resonances are expanded in powers of. It is shown that the zero-order terms correspond to the standard CAP method and that higher-order corrections are significant. The introduction of a convergence operator allows to control the convergence of the perturbation series. Due to the discretization of the continuum, the series are generally asymptotic. Finally, we relate the perturbation series to numerically convenient Taylor series. The theory is illustrated on two model examples.
