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The spectrum of the Hermitian Hamiltonian $H=p^2+V(x)$ is real and discreteif the potential $V(x)\to\infty$ as $x\to\pm\infty$. However, if $V(x)$ iscomplex and PT-symmetric, it is conjectured that, except in rare special cases,$V(x)$ must be analytic in order to have a real spectrum. This conjecture isdemonstrated by using the potential $V(x)=(ix)^a|x|^b$, where $a,b$ are real.

Conjecture on the analyticity of PT-symmetric potentials and the reality of their spectra