On the metric operator for the imaginary cubic oscillator

TL;DR

This paper investigates the spectral and basis properties of the PT-symmetric imaginary cubic oscillator, revealing the existence of a singular metric operator and spectral instabilities, which challenge traditional quantum interpretations.

Contribution

It demonstrates that the eigenvectors form a complete set but not a Riesz basis, and shows the metric operator is intrinsically singular, highlighting fundamental differences from self-adjoint Hamiltonians.

Findings

Eigenvectors are complete but do not form a Riesz basis.

Existence of a bounded, yet intrinsically singular, metric operator.

Presence of a non-trivial pseudospectrum indicating spectral instability.

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.