Completeness and Orthonormality in PT-symmetric Quantum Systems

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.

6 pages

References in corpus (14)

• Real Spectra in Non-Hermitian Hamiltonians Having PT Symmetry
• Complex Extension of Quantum Mechanics
• Pseudo-Hermiticity versus PT Symmetry: The necessary condition for the reality of the spectrum of a non-Hermitian Hamiltonian
• Pseudo-Hermiticity versus PT-Symmetry II: A complete characterizatio n of non-Hermitian Hamiltonians with a real spectrum
• Pseudo-Hermiticity versus PT-Symmetry III: Equivalence of pseudo-Her miticity and the presence of antilinear symmetries
• Exact PT-Symmetry Is Equivalent to Hermiticity
• Pseudo-Hermiticity and Generalized PT- and CPT-Symmetries
• Generalized Continuity Equation and Modified Normalization in PT-Symmetric Quantum Mechanics
• Space of State Vectors in PT Symmetrical Quantum Mechanics
• Pseudo-Hermiticity for a Class of Nondiagonalizable Hamiltonians
• Conjecture on the Interlacing of Zeros in Complex Sturm-Liouville Problems
• Finite-Dimensional PT-Symmetric Hamiltonians
• Solvable simulation of a double-well problem in PT symmetric quantum mechanics
• PT-symmetry and its spontaneous breakdown explained by anti-linearity

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