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Completeness and Orthonormality in PT-symmetric Quantum Systems

Stefan Weigert
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Abstract

Some PT-symmetric non-hermitean Hamiltonians have only real eigenvalues. There is numerical evidence that the associated PT-invariant energy eigenstates satisfy an unconventional completeness relation. An ad hoc scalar product among the states is positive definite only if a recently introduced `charge operator' is included in its definition. A simple derivation of the conjectured completeness and orthonormality relations is given. It exploits the fact that PT-symmetry provides an additional link between the eigenstates of the Hamiltonian and those of its adjoint, which form a dual pair of bases. The `charge operator' emerges naturally upon expressing the properties of the dual bases in terms of one basis only.

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Stefan Weigert. 2003-06-05. Completeness and Orthonormality in PT-symmetric Quantum Systems. https://doi.org/10.1103/physreva.68.062111

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