Is the CPT-norm always positive?

Details Is the CPT-norm always positive? Is the CPT-norm always positive?

2005-03-03

arxiv.org/abs/quant-ph/0503040v1

J. Phys. A: Math. Gen. 38 (2005) L249--L255

Physics

Quantum Physics

Scientific paper

10.1088/0305-4470/38/15/L02

We give an explicit example of an exactly solvable PT-symmetric Hamiltonian with the unbroken PT symmetry which has one eigenfunction with the zero PT-norm. The set of its eigenfunctions is not complete in corresponding Hilbert space and it is non-diagonalizable. In the case of a regular Sturm-Liouville problem any diagonalizable PT-symmetric Hamiltonian with the unbroken PT symmetry has a complete set of positive CPT-normalazable eigenfunctions. For non-diagonalizable Hamiltonians a complete set of CPT-normalazable functions is possible but the functions belonging to the root subspace corresponding to multiple zeros of the characteristic determinant are not eigenfunctions of the Hamiltonian anymore.

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