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

arXiv:2302.10609

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

Wei Yang

Published Feb 21, 2023• 0 citations
arXivHTMLPDF

Abstract

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.

DOI
10.48550/arxiv.2302.10609

Related papers

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

Yang Wei

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.

Journal of Mathematical Physics•Aug 1, 2023•1 citation

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

New set of exactly solvable complex potentials giving the real energies

Miloslav Znojil

We deform the real potential of Poeschl and Teller by a shift of its coordinate in imaginary direction. We show that the new model remains exactly solvable. Its bound states are constructed in closed form. Wave functions are complex and proportional to Jacobi polynomials. Some of them diverge in the Hermitian limit. In contrast, all their energies prove real and shift-independent. In this sense the lost Hermiticity of our family of Hamiltonians seems replaced by their accidental PT symmetry.

arXiv:quant-ph/9912079•Dec 16, 1999•1 citation

PT-symmetrized exact solution of the singular Eckart oscillator

Miloslav Znojil

The well known Eckart's singular s-wave potential is PT-symmetrically regularized and continued to the whole real line. The new model remains exactly solvable and its bound states remain proportional to Jacobi polynomials. Its real and discrete spectrum exhibits several unusual features.

•Dec 7, 1999•0 citations
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)+itanh⁡(λx)]V(x)=A[\sech(λx)+i \tanh(λx)]V(x)=A[\sech(λx)+itanh(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.