Completeness and Orthonormality in PT-symmetric Quantum Systems

Title	Completeness and Orthonormality in PT-symmetric Quantum Systems
Publication Type	Journal Article
Year of Publication	2003
Authors	Weigert S
Journal	Phys. Rev. A
Volume	68
Pagination	062111
Abstract	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.
URL	http://link.aps.org/abstract/PRA/v68/e062111
DOI	10.1103/PhysRevA.68.062111
E-print number	arXiv:quant-ph/0306040

Department of Mathematics