Seminário Análise Funcional e Aplicações 

04/11/2011, 15:00 — 16:00 — Sala P3.10, Pavilhão de Matemática
David Krejcirik, Nuclear Physics Institute of Academy of Sciences, ASCR, Czech Republic

Non-Hermitian operators in quantum theory and PT-symmetry

We give an introductory talk on a recent progress in the so-called "PT-symmetric quantum theory", in which the usual self-adjointness of observables requirement is replaced by their simultaneous Parity-Time invariance. The latter "often" implies that the spectrum is real and that the time evolution is unitary when reconsidered in a Hilbert space with appropriately changed inner product. The relevance of PT-symmetry has been suggested in various domains of physics, however, so far, there has been no experimental evidence proving that quantum systems defined by PT-symmetric Hamiltonians do exist in nature. In this talk, inter alia, we propose a simple PT-symmetric interpretation of a class of Sturm-Liouville operators with non-Hermitian Robin-type boundary conditions as a (physical) perfect-transmission scattering problem. Moreover, we establish closed integral-type formulae for similarity transformations relating the non-Hermitian operators with self-adjoint Hamiltonians, succeed in writing down the latter as a simple integro-differential operator and also find the associated "charge conjugation" operator, which plays the role of fundamental symmetry in a Krein-space reformulation of the problem.