Action-at-a-distance in a solvable quantum model

Miloslav Znojil

Among quantum systems with finite Hilbert space a specific role is played by systems controlled by non-Hermitian Hamiltonian matrices \(H\neq H^\dagger\) for which one has to upgrade the Hilbert-space metric by replacing the conventional form \(\Theta^{(Dirac)}=I\) of this metric by a suitable upgrade \(\Theta^{(non−Dirac)}\neq I\) such that the same Hamiltonian becomes self-adjoint in the new, upgraded Hilbert space, \(H=H\ddagger=\Theta^{−1}H^\dagger\Theta\). The problems only emerge in the context of scattering where the requirement of the unitarity was found to imply the necessity of a non-locality in the interaction, compensated by important technical benefits in the short-range-nonlocality cases. In the present paper we show that an why these technical benefits (i.e., basically, the recurrent-construction availability of closed-form Hermitizing metrics \(\Theta^{(non−Dirac)}\) can survive also in certain specific long-range-interaction models.
Quantum Physics (quant-ph)

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