A new solvable complex PT-symmetric potential

2015/02/28 by Zafar Ahmed, Dona Ghosh, D. C. Ghosh +1
Mathematics · Physics and Astronomy · #Mathematical physics #Nonlinear Waves and Solitons #Physics #Quantum Mechanics and Non-Hermitian Physics #Theoretical physics #math-ph #math.MP #quant-ph

paper · pdf · doi:10.1016/j.physleta.2015.04.032

published as Phys. Lett. A 379 (2015) 1639 · 12 pages, 5 Figures, Ref.[21] newly added, Appendix removed

openalex publication_date 2015/04/25 · arxiv created 2015/06/10 · arxiv updated 2015/06/11 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05

Abstract

We propose a new solvable one-dimensional complex PT-symmetric potential as V(x)= ig~ sgn(x)~ |1-exp(2|x|/a)| and study the spectrum of H=-d²/dx²+V(x). For smaller values of a,g <1, there is a finite number of real discrete eigenvalues. As a and g increase, there exist exceptional points (EPs), g_n (for fixed values of a) causing a scarcity of real discrete eigenvalues, but there exists at least one. We also show these real discrete eigenvalues as poles of reflection coefficient. We find that the energy-eigenstates ψ_n(x) satisfy (1): PTψ_n(x)=1 ψ_n(x) and (2): PTψ_En(x)=ψ_E^*n(x), for real and complex energy eigenvalues, respectively.

Citations

• Reflectionless potentials and symmetry
• sl(2, ) as a complex Lie algebra and the associated non-Hermitian Hamiltonians with real eigenvalues
• Making sense of non-Hermitian Hamiltonians
• -symmetry and its spontaneous breakdown explained by anti-linearity
• Real Spectra in Non-Hermitian Hamiltonians HavingPTSymmetry
• A PT-invariant potential with complex QES eigenvalues
• Normal Form of Antiunitary Operators
• Conjecture on the analyticity of \cal PT -symmetric potentials and the reality of their spectra