Quantum mechanics using two auxiliary inner products

The current applications of non-Hermitian but ${\cal PT}-$symmetric Hamiltonians $H$ cover several, mutually not too closely connected subdomains quantum physics. Mathematically, the split between open and closed systems can be characterized by respective triviality non-triviality an auxiliary inner-product metric $\Theta=\Theta(H)$. With our attention restricted to latter, mathematically more interesting unitary-evolution case we show that intuitive technically decisive simplification theory achieved via "additional" PCT}-$symmetry constraint upon given a deeper mathematical meaning introduction certain second inner product.