부산대학교 도서관

The authors outline a straightforward approach to study the quantum mechanical scattering due to complex potentials in space dimension. They investigate the consequences of imposing parity (P) time-reversal (T), and PT symmetries on the scattering matrix, and explore various local and nonlocal toy models. They also consider the relation between complete absorption for certain complex potentials and their symmetries. The approach pursued in this paper takes complex potentials as means for defining effective models typically arising in the phenomenological treatments of open quantum systems. It makes the implicit assumption that the usual $x$ operator represents the position observable. This makes the relevance of the results to the recent study of quasi-Hermitian scattering Hamiltonians (for example the model studied in [A. Mostafazadeh, J. Math. Phys. {\bf 46} (2005), no.~10, 102108, 15~pp.; MR2178580 (2006g:81210)]) rather obscure. This is because the latter Hamiltonians can define fundamental unitary quantum mechanical systems. This is achieved by defining the physical Hilbert space of the system using an appropriate generally nonstandard inner product. In such theories the position observable does not coincide with the usual $x$ operator, and the amplitude of the solution of the Schrödinger equation does not give the probability of the localization in physical space. Therefore one must be very careful when one uses terms like left-going and right-going waves, even for asymptotic scattering states. Similarly, the conserved current densities obtained by the authors cannot be associated with genuine (positive) probability densities. The unitary systems defined by such potentials may be equivalently described by a class of nonlocal real potentials for which a local conservation law for probabilities is not known.