The potential (iz)m generates real eigenvalues only, under symmetric rapid decay boundary conditions.

We consider the non-Hermitian eigenvalue problems -u″(z)±(iz)mu(z)=λu(z), m>=3, under every rapid decay boundary condition that is symmetric with respect to the imaginary axis in the complex z plane. We prove that the eigenvalues λ are all real and positive. [ABSTRACT FROM AUTHOR]