Quasi-hermitian lattices with imaginary zero-range interactions

We study a general class of \(\mathscr {PT}\)-symmetric tridiagonal quantum Hamiltonians with purely imaginary interaction term in the quasi-Hermitian representation. This general Hamiltonian encompasses many previously studied lattice models as special cases. We provide numerical results regarding domains of observability and exceptional points, and discuss the possibility of explicit construction of general metric operators (which in turn determine all the physical Hilbert spaces). The condition of computational simplicity for the metrics motivates the introduction of certain one-parametric special cases, which consequently admit closed-form extrapolation patterns of the low-dimensional results.