Real and complex discrete eigenvalues in an exactly solvable one-dimensional complex PT-invariant potential. (English) Zbl 0984.81043

Summary: The discrete eigenvalues of the complex PT-invariant potential \(V(x)=(-V_1\) sech \(x-iV_2\text{tanh}x) \text{sech} x, V_1>0\), 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.


81Q10 Selfadjoint operator theory in quantum theory, including spectral analysis
Full Text: DOI


[1] D. Bessis, unpublished (1992)
[2] Bender, C.M.; Boettcher, S., Phys. rev. lett., 80, 5243, (1998)
[3] Bender, C.M.; Boettcher, S.; Meisinger, P.N.; Bender, C.M.; Boettcher, S.; Jones, H.F.; Savage, V.M., J. math. phys., J. phys. A: math. gen., 32, 6771, (1999) · Zbl 0946.81076
[4] Bender, C.M.; Cooper, F.; Meisinger, P.N.; Savage, V.M., Phys. lett. A, 259, 224, (1999)
[5] Znojil, M., Phys. lett. A, 259, 220, (1999)
[6] Znojil, M.; Znojil, M., J. phys. A: math. gen., J. phys. A: math. gen., 33, L61, (2000)
[7] Bagchi, B.; Roychoudhury, R., J. phys. A: math. gen., 33, L1, (2000) · Zbl 0967.81016
[8] Khare, A.; Mandal, B.P., Phys. lett. A, 272, 53, (2000)
[9] Razavy, M., Am. J. phys., 48, 285, (1980)
[10] Dabrowwaska, J.W.; Khare, A.; Sukhatme, U.P., J. phys. A: math. gen., 21, L195, (1988)
[11] Hautot, A., J. math. phys., 14, 1320, (1973)
[12] Gradshteyn, I.S.; Ryzhik, I.M., Table of integrals, series and products, (1980), Academic Press New York, p. 838, (7.375.2) · Zbl 0521.33001
[13] Z. Ahmed, submitted
[14] Nieto, M.M., Phys. rev., 17, 1273, (1978)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.