An exactly solvable quantum-lattice model with a tunable degree of nonlocality

A non-Hermitian N-site-lattice Hamiltonian H with Laguerre-polynomial right eigenvectors and real energies is made self-adjoint in an ad hoc Hilbert space H-(S). The necessary physical inner products are defined via alternative k-parametric (2k + 1)-diagonal metrics Theta = Theta(k) constructed, in the closed form, at k = 0, k = 1, k = 2 and k = 3. The value of k is interpreted as a degree of nonlocality of the model.