Real and complex discrete eigenvalues in an exactly solvable one-dimensional complex PT-invariant potential

The discrete eigenvalues of the complex PT-invariant potential V(x) = (- V-1 sech.v - iV(2) tanh x) sech x, V-1 > 0. are shown to be only complex-conjugate pairs when \V-2\ > V-1 + 1/4. and real otherwise. The PT symmetry is spontaneously broken in the former and unbroken in the latter case. Using one more potential we find that when its real part is stronger than its imaginary part, all the eigenvalues are real, and they are mixed otherwise. (C) 2001 Elsevier Science B.V. All rights reserved.