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 $\psi$ is controlled, in Hilbert space $\mathcal{L}$, by a Hamiltonian $\mathfrak{h}$ which must be self-adjoint. In the recent, quasi-Hermitian'' reformulation of the theory one replaces $\mathfrak{h}$ by its isospectral but non-Hermitian avatar $H = Ω^{-1}\mathfrak{h}Ω$ with $Ω^\daggerΩ= Θ\neq I$. Although acting in another, manifestly unphysical Hilbert space ${\cal H}$, the amended Hamiltonian $H \neq H^\dagger$ 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 $\mathrm{\Theta }$ become available in closed form.