Factorized Hilbert-space metrics and non-commutative quasi-Hermitian observables

Abstract

It is well known that an (in general, non-commutative) set of non-Hermitian operators $\Lambda_j$ with real eigenvalues need not necessarily represent observables. We describe a specific class of quantum models in which these operators plus the underlying physical Hilbert-space metric $\Theta$ are all represented in terms of an auxiliary operator $(N+1)-$plet $Z_k$, $k=0,1,\ldots,N$. Our formalism degenerates to the ${\cal PT}-$symmetric quantum mechanics at $N=2$, with metric $\Theta=Z_2Z_1$, parity ${\cal P}=Z_2$, charge ${\cal C}=Z_1$ and Hamiltonian $H=Z_0$.