Completeness and Orthonormality in PT-symmetric Quantum Systems

Details Completeness and Orthonormality in PT-symmetric Quantum Systems Completeness and Orthonormality in PT-symmetric Quantum Systems

2003-06-05

arxiv.org/abs/quant-ph/0306040v1

Phys. Rev. A 68 (2003) 062111

Physics

Quantum Physics

6 pages

Scientific paper

10.1103/PhysRevA.68.062111

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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