Let us first warn the reader that although the authors of current
textbooks on quantum mechanics rarely fail to mention several
exactly solvable illustrative examples using a one-dimensional
real potential (like linear harmonic oscillator), I am not aware
of a textbook where some real spectrum of bound states would be
discussed as generated by a potential which wouldn't be real.
Still, such an appealing possibility exists and has been described
in a number of papers. In a gap-filling mood G. Levai performs one
more step and, in an interesting completion of his older results
[27] he studies (separable) partial differential Schroedinger
equations with complex solvable potentials and real spectra in two
and three dimensions. He shows that many new and interesting
phenomenological spectra given by closed formulae can be obtained
in this manner. In his analysis he constructively demonstrates
that an underlying and, in one dimension, successfully tested
heuristic principle of choosing these potentials in the so called
PT-symmetric form (whatever the PT-symmetry means) also pays off
in more dimensions. In two illustrative examples the polar
ordinary Schroedinger equation is made solvable via the Scarf I
and Rosen-Morse I choice of the polar-angular part of the
potential.
MR2455813 Lévai, G. $\scr{PT}$ symmetry and its spontaneous
breakdown in three dimensions. J. Phys. A: Math. Gen. 41 (2008),
no. 24, 244015, 10 pp. 81Q10