  In conventional Schr\"{o}dinger representation the unitarity of the evolution
of bound states is guaranteed by the Hermiticity of the Hamiltonian. A
non-unitary isospectral simplification of the Hamiltonian, $\mathfrak{h} \to
H=\Omega\,\mathfrak{h}\,\Omega \neq H^\dagger$ induces the change ${\cal L} \to
{\cal K}$ of the Hilbert space of states, reflected by the loss of the
Hermiticity of $H\neq H^\dagger$. In such a reformulation of the theory the
introduction of an {\it ad hoc} inner-product metric reconverts ${\cal K}$ into
the third, correct physical Hilbert space ${\cal H}$, unitarily equivalent to
${\cal L}$. The situation encountered, typically, in ${\cal PT}-$symmetric or
relativistic quantum mechanics is shown more complicated after an inclusion of
perturbations. The formulation and solution of the problem are presented. Some
of the consequences relevant, e.g., in the analysis of stability are discussed.
