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Eight exactly solvable complex potentials in Bender-Boettcher quantum mechanics. (English) Zbl 0977.81011
Slovák, Jan (ed.) et al., The proceedings of the 20th winter school “Geometry and physics”, Srní, Czech Republic, January 15-22, 2000. Palermo: Circolo Matematico di Palermo, Suppl. Rend. Circ. Mat. Palermo, II. Ser. 66, 213-218 (2001).
This is a readable review of recent work on non-Hermitian bound state problems with complex potentials. A particular example is the generalization of the harmonic oscillator with the potentials: \[ V(x)=\frac{\omega^2}2\,\left(x-\frac{2i\beta}{\omega}\right)^2-\frac{\omega}{2}. \] Other examples include complex generalizations of the Morse potential, the spiked radial harmonic potential, the Kratzer-Coulomb potential, the Rosen Morse oscillator and others. Instead of demanding Hermiticity \(H=H^*\) the condition required is \(H=PTHPT\) where \(P\) changes the parity and \(T\) transforms \(i\) to \(-i\).
For the entire collection see [Zbl 0961.00020].

81Q05 Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics
81U15 Exactly and quasi-solvable systems arising in quantum theory