In the conventional Schr\"{ o}dinger's formulation of quantum mechanics the unitary evolution of a state $\psi$ is controlled, in Hilbert space ${ \cal L}$, 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 $H = \Omega^{ -1}\mathfrak{ h}\Omega$ with $\Omega^\dagger\Omega = \Theta \neq I$. 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 $\Theta$. 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 $\Theta$ become available in closed form.
Categories: quant-ph