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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 $-i$.
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