Completeness and orthonormality in PT-symmetric quantum systems

TY - JOUR

T1 - Completeness and orthonormality in PT-symmetric quantum systems

AU - Weigert, S.

N1 - © 2003 The American Physical Society. Reproduced in accordance with the publisher's self-archiving policy.

PY - 2003/12

Y1 - 2003/12

N2 - Some PT-symmetric non-Hermitian 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 a link between the eigenstates of the Hamiltonian and those of its adjoint, forming a dual pair of bases. The charge operator emerges naturally upon expressing the properties of the dual bases in terms of one basis only, and it is shown to be a function of the Hamiltonian.

AB - Some PT-symmetric non-Hermitian 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 a link between the eigenstates of the Hamiltonian and those of its adjoint, forming a dual pair of bases. The charge operator emerges naturally upon expressing the properties of the dual bases in terms of one basis only, and it is shown to be a function of the Hamiltonian.

U2 - 10.1103/PhysRevA.68.062111

DO - 10.1103/PhysRevA.68.062111

M3 - Article

SN - 1050-2947

VL - 68

SP - art no. 062111

JO - Physical Review A

JF - Physical Review A

IS - 6

ER -