Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Journal of Mathematical Physics

Exactly solvable complex PT symmetry potential A[sech(λx) + i tanh(λx)]

Yang Wei

Published Aug 1, 2023• 1 citation
Publisher

Abstract

We obtained the exact solutions of the PT symmetric potential V(x) = A[sech(λx) + i tanh(λx)] and found that this system has no bound-state. The PT symmetric potential was first studied in this article, and the handedness effect is shown from reflection coefficients. As the asymptotically non-vanishing imaginary potential component, when the direction of the incident wave is opposite, the transmission coefficient will emerge a complex phase factor.

DOI
10.1063/5.0150043

Related papers

Exactly solvable complex PT symmetry potential $A[\mathrm{sech}(λx) + i\tanh(λx)]$

Wei Yang

We obtained the exactly solutions of the $\mathcal{PT}$ symmetric potential $V(x)=A[\sech(λx)+i \tanh(λx)]$, and found this system has no bound-state. which $\mathcal{PT}$ symmetric potential was first studied in this article, and the handedness effect is showed from reflection coefficients. As the asymptotically non-vanishing imaginary potential component, when the direction of the incident wave is opposite, that the transmission coefficient will emerge a complex phase factor.

arXiv:2302.10609•Feb 21, 2023•0 citations

The {\cal PT} -symmetric Rosen–Morse II potential: effects of the asymptotically non-vanishing imaginary potential component

GÉZA LÉVAI, E. Magyari

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.

Journal of Physics A Mathematical and Theoretical•Apr 22, 2009•33 citations

PT symmetric form of the Hulthen potential

Miloslav Znojil

The fourth, missing example of an exactly solvable complex potential with PT symmetry V(x) = [V(-x)]^* defined on a bent contour and leading, at the real energies, to the Jacobi polynomial wave functions is found in a generalized Hulthen interaction.

arXiv:math-ph/0002017•Feb 7, 2000•0 citations

Scattering in PT-symmetric quantum mechanics

F. Cannata, J.-P. Dedonder, Alberto Ventura

A general formalism is worked out for the description of one-dimensional scattering in non-hermitian quantum mechanics and constraints on transmission and reflection coefficients are derived in the cases of P, T, or PT invariance of the Hamiltonian. Applications to some solvable PT-symmetric potentials are shown in detail. Our main original results concern the association of reflectionless potentials with asymptotic exact PT symmetry and the peculiarities of separable kernels of non-local potentials in connection with hermiticity, T invariance and PT invariance.

arXiv:quant-ph/0606129•Jun 15, 2006•73 citations

Algebraic and scattering aspects of a 𝒫𝒯-symmetric solvable potential

GÉZA LÉVAI, F. Cannata, Alberto Ventura

We study a particular solvable potential and analyse the effect of 𝒫𝒯 symmetry on its bound state as well as scattering solutions. We determine the transmission and reflection coefficients for the 𝒫𝒯-symmetric case and also formulate the problem in terms of an SU (1,1) potential group, which allows unified treatment of the discrete and the continuous spectra in a natural way. We find that (bound and scattering) states of the 𝒫𝒯-symmetric problem supply a basis for the unitary irreducible representations of the SU (1,1) potential group, and this gives a straightforward group theoretical interpretation of the fact that the (complex) 𝒫𝒯-invariant potential has a real energy spectrum.

Journal of Physics A Mathematical and General•Jan 18, 2001•82 citations

PT-/non-PT-symmetric and non-Hermitian Hellmann potential: approximate bound and scattering states with anyℓ-values

Altuğ Arda, Ramazan Sever

We investigate the approximate bound state solutions of the Schrödinger equation for the PT-/non-PT-symmetric and non Hermitian Hellmann potential. Exact energy eigenvalues and corresponding normalized wave functions are obtained. Numerical values of energy eigenvalues for the bound states are compared with the ones obtained before. Scattering state solutions are also studied. Phase shifts of the potential are written in terms of the angular momentum quantum number $\ell$.

arXiv:1409.0518•Physica Scripta•Aug 28, 2014•2 citations
We obtained the exact solutions of the PT symmetric potential V(x) = A[sech(λx) + i tanh(λx)] and found that this system has no bound-state. The PT symmetric potential was first studied in this article, and the handedness effect is shown from reflection coefficients. As the asymptotically non-vanishing imaginary potential component, when the direction of the incident wave is opposite, the transmission coefficient will emerge a complex phase factor.
Exactly solvable complex PT symmetry potential A[sech(λx)+itanh⁡(λx)]A[\mathrm{sech}(λx) + i\tanh(λx)]A[sech(λx)+itanh(λx)]
We obtained the exactly solutions of the PT\mathcal{PT}PT symmetric potential $V(x)=A[\sech(λx)+i \tanh(λx)]$, and found this system has no bound-state. which PT\mathcal{PT}PT symmetric potential was first studied in this article, and the handedness effect is showed from reflection coefficients. As the asymptotically non-vanishing imaginary potential component, when the direction of the incident wave is opposite, that the transmission coefficient will emerge a complex phase factor.
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.
A general formalism is worked out for the description of one-dimensional scattering in non-hermitian quantum mechanics and constraints on transmission and reflection coefficients are derived in the cases of P, T, or PT invariance of the Hamiltonian. Applications to some solvable PT-symmetric potentials are shown in detail. Our main original results concern the association of reflectionless potentials with asymptotic exact PT symmetry and the peculiarities of separable kernels of non-local potentials in connection with hermiticity, T invariance and PT invariance.
We study a particular solvable potential and analyse the effect of 𝒫𝒯 symmetry on its bound state as well as scattering solutions. We determine the transmission and reflection coefficients for the 𝒫𝒯-symmetric case and also formulate the problem in terms of an SU (1,1) potential group, which allows unified treatment of the discrete and the continuous spectra in a natural way. We find that (bound and scattering) states of the 𝒫𝒯-symmetric problem supply a basis for the unitary irreducible representations of the SU (1,1) potential group, and this gives a straightforward group theoretical interpretation of the fact that the (complex) 𝒫𝒯-invariant potential has a real energy spectrum.
We investigate the approximate bound state solutions of the Schrödinger equation for the PT-/non-PT-symmetric and non Hermitian Hellmann potential. Exact energy eigenvalues and corresponding normalized wave functions are obtained. Numerical values of energy eigenvalues for the bound states are compared with the ones obtained before. Scattering state solutions are also studied. Phase shifts of the potential are written in terms of the angular momentum quantum number ℓ\ellℓ.