THE COUPLED-CLUSTER APPROACH TO QUANTUM MANY-BODY PROBLEM IN A THREE-HILBERT-SPACE REINTERPRETATION

Raymond F. Bishop, Miloslav Znojil

Abstract


The quantum many-body bound-state problem in its computationally successful coupled cluster method (CCM) representation is reconsidered. In conventional practice one factorizes the groundstate wave functions |Ψ) = eS |Φ) which live in the “physical” Hilbert space H(P) using an elementary ansatz for |Φi plus a formal expansion of S in an operator basis of multi-configurational creation operators C+. In our paper a reinterpretation of the method is proposed. Using parallels between the CCM and the so called quasi-Hermitian, alias three-Hilbert-space (THS), quantum mechanics, the CCM transition from the known microscopic Hamiltonian (denoted by usual symbol H), which is self-adjoint in H(P), to its effective lower-case isospectral avatar h = e−SHeS, is assigned a THS interpretation. In the opposite direction, a THS-prescribed, non-CCM, innovative reinstallation of Hermiticity is shown to be possible for the CCM effective Hamiltonian h, which only appears manifestly non-Hermitian in its own (“friendly”) Hilbert space H(F). This goal is achieved via an ad hoc amendment of the inner product in H(F), thereby yielding the third (“standard”) Hilbert space H(S). Due to the resulting exact unitary equivalence between the first and third spaces, H(F)H(S), the indistinguishability of predictions calculated in these alternative physical frameworks is guaranteed.

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ISSN 1210-2709 (Print)
ISSN 1805-2363 (Online)
Published by the Czech Technical University in Prague