Complete Set of Inner Products for a Discrete PT-symmetric Square-well Hamiltonian

A discrete $N-$point Runge-Kutta version $H^{(N)}(λ)$ of one of the simplest non-Hermitian square-well Hamiltonians with real spectrum is studied. A complete set of its possible hermitizations (i.e., of the eligible metrics $Θ^{(N)}(λ)$ defining its non-equivalent physical Hilbert spaces of states) is constructed, in closed form, for any coupling $λ\in (-1,1)$ and any matrix dimension $N$.

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