The Large −g Observability of the Low-Lying Energies in the Strongly Singular Potentials V (x) = x 2 + g 2/x 6 after their 𝓟T −symmetric Regularization
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The elementary quadratic plus inverse sextic interaction V (x) = x 2 + g 2/x 6 containing a strongly singular repulsive core in the origin is made regular by a complex shift of coordinate x = s −iε. The shift ε > 0 is fixed while the value of s is kept real and potentially observable, s ∈ (−∞, ∞). The low-lying energies of bound states are found in closed form for the large couplings g ≫ 1. Within the asymptotically vanishing 𝒪(g −1/4) error bars these energies are real so that the time-evolution of the system may be expected unitary in an ad hoc physical Hilbert space.
KeywordsQuantum evolution Triple-Hilbert-space picture Strongly singular forces Regularization by complexification Strong-coupling dynamical regime Unitarity
Discussions of the subject with Roberto Tateo and with several other colleagues are gratefully appreciated.
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