Quasi-hermitian quantum mechanics and a new class of user-friendly matrix hamiltonians

In the conventional Schrödinger's formulation of quantum mechanics the unitary evolution of a state ψ is controlled, in Hilbert space ${\mathcal{L}}$, by a Hamiltonian ɧ which must be self-adjoint. In the recent, "quasi-Hermitian" reformulation of the theory one replaces ɧ by its isospectral but non-Hermitian avatar H = Ω⁻¹ 𝖍Ω with Ω^†Ω = Θ ≠ I. Although acting in another, manifestly unphysical Hilbert space ${\mathcal{H}}$, the amended Hamiltonian H ≠ H^† can be perceived as self-adjoint with respect to the amended inner-product metric Θ. In our paper motivated by a generic technical "user-unfriendliness" of the non-Hermiticity of H we introduce and describe a specific new family of Hamiltonians H for which the metrics Θ become available in closed form.