  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.
