This paper explains the systematics of the generation of families of spectra for the PT-symmetric quantum-mechanical Hamiltonians \(H=p^2+x^2(ix)^\epsilon\), \(H=p^2+(x^2)^\delta\), and \(H=p^2-(x^2)^\mu\). In addition, it contrasts the results obtained with those found for a bosonic scalar field theory, in particular in one dimension, highlighting the similarities and differences to the quantum-mechanical case. It is shown that the number of families of spectra can be deduced from the number of noncontiguous pairs of Stokes’ wedges that display PT-symmetry. To do so, simple arguments that use the WKB approximation are employed, and these imply that the eigenvalues are real. However, definitive results are in most cases presently only obtainable numerically, and not all eigenvalues in each family may be real. Within the approximations used, it is illustrated that the difference between the quantum-mechanical and the field-theoretical cases lies in the number of accessible regions in which the eigenfunctions decay exponentially. This paper reviews and implements well-known techniques in complex analysis and PT-symmetric quantum theory.

http://arxiv.org/abs/1204.4599
High Energy Physics – Theory (hep-th); Mathematical Physics (math-ph); Quantum Physics (quant-ph)