PT symmetry and shape invariance for a potential well with a barrier. (English) Zbl 0985.81031
Summary: We construct an exponential-type PT-symmetric potential, which includes the PT-symmetric versions of the Rosen-Morse well and Scarf potential and the complex PT-invariant potential well \(V(x)=q_2\tan h^2\alpha x+i(q_1/2)\text{sech}\alpha x \tan h\alpha x+q_0, q_2>0\). The discrete energy eigenvalues of the latter complex potential are shown to be real when \(|q_1|\leqslant\alpha^2/2+2q_2\), while they are complex conjugate pairs if \(|q_1|>\alpha^2/2+2q_2\). The PT symmetry is unbroken in the former case and spontaneously broken in the latter case.
MSC:
| 81Q15 | Perturbation theories for operators and differential equations in quantum theory |
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