Two PT-symmetric potentials are compared, which possess asymptotically finite imaginary components: the PT-symmetric Rosen-Morse II and the finite PT-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 asymptotically non-vanishing imaginary potential components. Here the V(x)= γδ(x)+ i2Λ sgn(x) 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.
Abstract The -symmetric version of the one-dimensional finite square well potential is constructed by supplementing the real finite square well potential with constant imaginary components outside the well. This study is motivated by the unusual features of the -symmetric Rosen–Morse II potential, which has an imaginary component with similar asymptotics. The exponentially vanishing (i.e. normalizable) solutions are constructed, after determining the corresponding energy eigenvalues from the zeros of a transcendental equation. It is found that only real energy eigenvalues are allowed, similarly to the case of the Rosen–Morse II potential. Transmission and reflection coefficients are determined and are found to exhibit handedness. It is shown that due to the non-vanishing imaginary potential component, bound states correspond to the zeros of the reflection coefficient. Similarities and differences with respect to the real finite square well and the -symmetric Rosen–Morse II potential are discussed.
Bound and scattering solutions of the -symmetric Rosen–Morse II potential are investigated. The energy eigenvalues and the corresponding wavefunctions are written in a closed analytic form, and it is shown that this potential always supports at least one bound state. It is found that with increasing non-Hermiticity the real bound-state energy spectrum does not turn into complex conjugate pairs, i.e. the spontaneous breakdown of symmetry does not occur, rather the energy eigenvalues remain real and shift to positive values. Closed expression is found for the pseudo-norm of the bound states, and its sign is found to follow the (−1)n rule. Similarly to the known scattering examples, the reflection coefficients exhibit a handedness effect, while the transmission coefficient picks up a complex phase factor when the direction of the incoming wave is reversed. It is argued that the unusual findings might be caused by the asymptotically non-vanishing, though finite imaginary potential component. Comparison with the real Rosen–Morse II potential is also made.
Schroedinger equation with imaginary PT-symmetric potential
V(x)=ix3 is studied using the numerical discretization methods in both the coordinate and momentum representations. In the former case our results confirm that the model generates an infinite number of bound states with real energies. In the latter case the differential equation is of the third order and a square-well, solvable approximation of kinetic energy is recommended and discussed. One finds that in the strong-coupling limit, the exact PT-symmetric solutions converge to their Hermitian predecessors.
One-dimensional PT-symmetric quantum-mechanical Hamiltonians having continuous spectra are studied. The Hamiltonians considered have the form
H=p2+V(x), where
V(x) is odd in
x, pure imaginary, and vanishes as
∣x∣→∞. Five PT-symmetric potentials are studied: the Scarf-II potential
V1(x)=iA1sech(x)tanh(x), which decays exponentially for large
∣x∣; the rational potentials
V2(x)=iA2x/(1+x4) and
V3(x)=iA3x/(1+∣x∣3), which decay algebraically for large
∣x∣; the step-function potential
V4(x)=iA4sgn(x)θ(2.5−∣x∣), which has compact support; the regulated Coulomb potential
V5(x)=iA5x/(1+x2), which decays slowly as
∣x∣→∞ and may be viewed as a long-range potential. The real parameters
An measure the strengths of these potentials. Numerical techniques for solving the time-independent Schrödinger eigenvalue problems associated with these potentials reveal that the spectra of the corresponding Hamiltonians exhibit universal properties. In general, the eigenvalues are partly real and partly complex. The real eigenvalues form the continuous part of the spectrum and the complex eigenvalues form the discrete part of the spectrum. The real eigenvalues range continuously in value from
0 to
+∞. The complex eigenvalues occur in discrete complex-conjugate pairs and for
Vn(x) (
1≤n≤4) the number of these pairs is finite and increases as the value of the strength parameter
An increases. However, for
V5(x) there is an {\it infinite} sequence of discrete eigenvalues with a limit point at the origin. This sequence is complex, but it is similar to the Balmer series for the hydrogen atom because it has inverse-square convergence.