Construction of a unique metric in quasi-Hermitian quantum mechanics: Nonexistence of the charge operator in a 2x2 matrix model

For a specific exactly solvable 2 x 2 matrix model with a PT-symmetric Hamiltonian possessing a real spectrum, we construct all the eligible physical metrics Theta > 0 and show that none of them admits a factorization Theta = CP in terms of an involutive charge operator C. Alternative ways of restricting the physical metric to a unique form are briefly discussed. (c) 2006 Elsevier B.V. All rights reserved.