Petr Siegl, David Krejcirik

We show that the eigenvectors of the PT -symmetric imaginary cubic oscillator are complete, but do not form a Riesz basis. This results in the existence of a bounded metric operator having intrinsic singularity reflected in the inevitable unboundedness of the inverse. Moreover, the existence of nontrivial pseudospectrum is observed. In other words, there is no quantum-mechanical Hamiltonian associated with it via bounded and boundedly invertible similarity transformations. These results open new directions in physical interpretation of PT -symmetric models with intrinsically singular metric, since their properties are essentially different with respect to self-adjoint Hamiltonians, for instance, due to spectral instabilities.

http://arxiv.org/abs/1208.1866

Quantum Physics (quant-ph); Mathematical Physics (math-ph)

Physical Review D **86**, 121702 (2012)