Abstract
Two <em>PT</em>-symmetric potentials are compared, which possess asymptotically finite imaginary components: the <em>PT</em>-symmetric Rosen-Morse II and the finite <em>PT</em>-symmetric square well potentials. Despite their different mathematical structure, their shape is rather similar, and this fact leads to similarities in their physical characteristics. Their bound-state energy spectrum was found to be purely real, an this finding was attributed to their <br />asymptotically non-vanishing imaginary potential components. Here the <em>V(x</em>)= <em>γδ</em>(<em>x</em>)+ i2Λ sgn(<em>x</em>) potential is discussed, which can be obtained as the common limit of the two other potentials. The energy spectrum, the bound-state wave functions and the transmission and reflection coefficients are studied in the respective limits, and the results are compared.
Highlights
The introduction of PT -symmetric quantum mechanics [1] gave strong impetus to the investigation of non-hermitian quantum mechanical systems
Note that PT symmetry, and in particular, the asymptotically non-vanishing potential component has strong influence on the asymptotic properties of the wave functions, and this fact manifests itself in the structure of the transmission and reflection coefficients too, in accordance with the findings of [22]
We investigated the PT -symmetric Rosen–Morse II and finite square well potentials in the limit when their real even potential component turns into the Dirac delta, while their imaginary odd component tend to the sign function, respectively
Summary
The introduction of PT -symmetric quantum mechanics [1] gave strong impetus to the investigation of non-hermitian quantum mechanical systems (for a review, see [2]). In the case of the Scarf II potential the imaginary potential component vanishes asymptotically, while in the case of the Rosen–Morse II potential it is the i tanh(x) function, reaching finite values for x → ±∞ In the former case the breakdown of PT symmetry can occur [15], while in the latter the discrete energy spectrum is purely real [10]. This latter finding was later proven for all three PII-class shape-invariant potentials (Rosen–Morse I, II, Eckart) using a thorough analysis of PT -symmetric Natanzon-class potentials [7].
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