PT-symmetric Quantum Mechanics, the hydrodynamic Squire equation and UV-IR-duality

Some facts about the spectrum of a PT-symmetric quantum mechanical (PTSQM) toy model with potential V(x)=Gx²(ix)^ν in a box x∈[-L,L] are presented for the parameter region ν∈[-2,0]. The corresponding Hamiltonian is selfadjoint in an appropriately chosen Krein space and for ν=-1 the spectral problem maps into that of the hydrodynamic Squire equation. It is shown that in the limit L→∞ a spectral singularity occurs and that the PTSQM ⇄ Squire mapping can be interpreted as a special type of strong-coupling ⇄ weak-coupling (UV-IR) duality. Finally, the system behavior in the vicinity of a spectral triple point is sketched.
partially based on:
J. Math. Phys. 46, (2005), 063504, math-ph/0501069.
Czech. J. Phys. 55, (2005), 1099-1106, math-ph/0506021.