Generalized PT-symmetric operators acting an a Hilbert space $mathfrak{H}$ are defined and investigated. The case of PT-symmetric extensions of a symmetric operator $S$ is investigated in detail. The possible application of the Lax-Phillips scatterin
g methods to the investigation of PT-symmetric operators is illustrated by considering the case of 0-perturbed operators.
We investigate the equivalence of quantum mixed states under local unitary transformations. For a class of rank-two mixed states, a sufficient and necessary condition of local equivalence is obtained by giving a complete set of invariants under local
unitary transformations, such that two states in this class are locally equivalent if and only if all these invariants have equal values for them.
