2012/08/31 by Petr Siegl, David Krejcirik, David Krejčiřı́k · 3 citations
Mathematics · Physics and Astronomy · #Bounded function #Eigenvalues and eigenvectors #Geometry #Hamiltonian (control theory) #Inverse #Invertible matrix #Mathematical analysis #Mathematical physics #Mathematics #Metric (unit) #Nonlinear Photonic Systems #Operator (biology) #Physics #Pure mathematics #Quantum #Quantum Mechanics and Non-Hermitian Physics #Quantum chaos and dynamical systems #Quantum mechanics #Singularity #The Imaginary #math-ph #math.MP #quant-ph
paper · pdf · doi:10.1103/physrevd.86.121702
published as Phys. Rev. D 86 (2012), 121702(R) · 7 pages; completely rewritten, new results
openalex publication_date 2012/12/04 · arxiv created 2012/12/13 · arxiv updated 2015/06/11 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
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.