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arXiv · 1208.1866

On the metric operator for the imaginary cubic oscillator

Petr Siegl·David Krejcirik
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Abstract

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 non-trivial 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.

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Petr Siegl, David Krejcirik. 2012-12-13. On the metric operator for the imaginary cubic oscillator. https://doi.org/10.1103/physrevd.86.121702

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