We give an introduction to the study of one particular class of non-self-adjoint operators, namely PF-symmetric ones. We explain briefly the physical motivation and describe the classes of operators that are considered. We explain relations between the operator classes, namely their non-equivalence, and mention open problems.

        

In the second part, we focus on the similarity to self-adjoint operators. On the positive side, we present results on one-dimensional Schrödinger-type operators in a bounded interval with non-self-adjoint Robin-type boundary conditions. Using functional calculus, closed formulas for the similarity transformation and the similar self-adjoint operator are derived in particular cases. On the other hand, we analyse the imaginary cubic oscillator, which, although being PF -symmetric and possessing real spectrum, is not similar to any self-adjoint operator. The argument is based on known semiclassical results.

        

1. P. Siegl: The non-equivalence of pseudo-Hermiticity and presence of antilinear symmetry, PRAMANA-Journal of Physics, Vol. 73, No. 2, 279-287,

2. D. Krejcirík, P. Siegl and J. Zelezný: On the similarity of Sturm-Liouville operators with non-Hermitian boundary conditions to self-adjoint and normal operators, Complex Analysis and Operator Theory, to appear,

3. P. Siegl and D. Krejcirík: On the metric operator for imaginary cubic oscillator, Physical Review D, to appear.