arXiv:1411.3231 · Physics Letters A

Real discrete spectrum in the complex non-PT-symmetric Scarf II potential

Zafar Ahmed, Joseph Amal Nathan

Published Jan 14, 2015• 9 citations

arXiv HTML PDF

DOI: 10.1016/j.physleta.2015.01.004

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arXiv:1411.3231 · Physics Letters A

Real discrete spectrum in the complex non-PT-symmetric Scarf II potential

Zafar Ahmed, Joseph Amal Nathan

Published Jan 14, 2015• 9 citations

arXiv HTML PDF

DOI: 10.1016/j.physleta.2015.01.004

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New features of scattering from a one-dimensional non-Hermitian (complex) potential

Zafar Ahmed

For complex one-dimensional potentials, we propose the asymmetry of both reflectivity and transmitivity under time reversal: R ( − k ) ≠ R ( k ) and T ( − k ) ≠ T ( k ), unless the potentials are real or PT-symmetric. For complex PT-symmetric scattering potentials, we propose that R left ( − k ) = R right ( k ) and T ( − k ) = T ( k ). So far, the spectral singularities (SS) of a one-dimensional non-Hermitian scattering potential are witnessed/conjectured to be at most 1. We present a new non-Hermitian parametrization of the Scarf II potential to reveal its four new features. Firstly, it displays the just acclaimed (in)variances. Secondly, it can support two spectral singularities at two pre-assigned real energies ( E * = α 2 , β 2 ) either in T ( k ) or in T ( − k ), when αβ > 0. Thirdly, when αβ < 0 it possesses one SS in T ( k ) and the other in T ( − k ). Fourthly, when the potential becomes PT-symmetric [(α + β) = 0], we obtain T ( k ) = T ( − k ), it possesses a unique SS at E = α 2 in both T ( − k ) and T ( k ). Lastly, for completeness, when α = iγ and β = iδ there are no SS, instead we get two real energies −γ 2 and −δ 2 of the complex PT-symmetric Scarf II belonging to the two well-known branches of discrete bound-state eigenvalues. We find them as E + M = −(γ − M ) 2 and E − N = −(δ − N ) 2 ; M ( N ) = 0, 1, 2, ... with 0 ⩽ M ( N ) < γ(δ).For complex one-dimensional potentials, we propose the asymmetry of both reflectivity and transmitivity under time reversal: R ( − k ) ≠ R ( k ) and T ( − k ) ≠ T ( k ), unless the potentials are real or PT-symmetric. For complex PT-symmetric scattering potentials, we propose that R left ( − k ) = R right ( k ) and T ( − k ) = T ( k ). So far, the spectral singularities (SS) of a one-dimensional non-Hermitian scattering potential are witnessed/conjectured to be at most 1. We present a new non-Hermitian parametrization of the Scarf II potential to reveal its four new features. Firstly, it displays the just acclaimed (in)variances. Secondly, it can support two spectral singularities at two pre-assigned real energies ( E * = α 2 , β 2 ) either in T ( k ) or in T ( − k ), when αβ > 0. Thirdly, when αβ < 0 it possesses one SS in T ( k ) and the other in T ( − k ). Fourthly, when the potential becomes PT-symmetric [(α + β) = 0], we obtain T ( k ) = T ( − k ), it possesses a unique SS at E = α 2 in both T ( − k ) and T ( k ). Lastly, for completeness, when α = iγ and β = iδ there are no SS, instead we get two real energies −γ 2 and −δ 2 of the complex PT-symmetric Scarf II belonging to the two well-known branches of discrete bound-state eigenvalues. We find them as E + M = −(γ − M ) 2 and E − N = −(δ − N ) 2 ; M ( N ) = 0, 1, 2, ... with 0 ⩽ M ( N ) < γ(δ).

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One-dimensional PT-symmetric quantum-mechanical Hamiltonians having continuous spectra are studied. The Hamiltonians considered have the form $H=p^2+V(x)$ , where $V(x)$ is odd in $x$ , pure imaginary, and vanishes as $|x|\to\infty$ . Five PT-symmetric potentials are studied: the Scarf-II potential $V_1(x)=iA_1\,{\rm sech}(x)\tanh(x)$ , which decays exponentially for large $|x|$ ; the rational potentials $V_2(x)=iA_2\,x/(1+x^4)$ and $V_3(x)=iA_3\,x/(1+|x|^3)$ , which decay algebraically for large $|x|$ ; the step-function potential $V_4(x)=iA_4\,{\rm sgn}(x)θ(2.5-|x|)$ , which has compact support; the regulated Coulomb potential $V_5(x)=iA_5\,x/(1+x^2)$ , which decays slowly as $|x|\to\infty$ and may be viewed as a long-range potential. The real parameters $A_n$ 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 $+\infty$ . The complex eigenvalues occur in discrete complex-conjugate pairs and for $V_n(x)$ ( $1\leq n\leq4$ ) the number of these pairs is finite and increases as the value of the strength parameter $A_n$ increases. However, for $V_5(x)$ 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.One-dimensional PT-symmetric quantum-mechanical Hamiltonians having continuous spectra are studied. The Hamiltonians considered have the form $H=p^2+V(x)$, where $V(x)$ is odd in $x$, pure imaginary, and vanishes as $|x|\to\infty$. Five PT-symmetric potentials are studied: the Scarf-II potential $V_1(x)=iA_1\,{\rm sech}(x)\tanh(x)$, which decays exponentially for large $|x|$; the rational potentials $V_2(x)=iA_2\,x/(1+x^4)$ and $V_3(x)=iA_3\,x/(1+|x|^3)$, which decay algebraically for large $|x|$; the step-function potential $V_4(x)=iA_4\,{\rm sgn}(x)θ(2.5-|x|)$, which has compact support; the regulated Coulomb potential $V_5(x)=iA_5\,x/(1+x^2)$, which decays slowly as $|x|\to\infty$ and may be viewed as a long-range potential. The real parameters $A_n$ 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 $+\infty$. The complex eigenvalues occur in discrete complex-conjugate pairs and for $V_n(x)$ ($1\leq n\leq4$) the number of these pairs is finite and increases as the value of the strength parameter $A_n$ increases. However, for $V_5(x)$ 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.

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Zafar Ahmed, D. C. Ghosh, Joseph Amal Nathan

arXiv:1502.04838Physics Letters AApr 25, 201512 citations

One-dimensional PT-symmetric quantum-mechanical Hamiltonians having continuous spectra are studied. The Hamiltonians considered have the form

H=p^2+V(x)

, where

V(x)

is odd in

x

, pure imaginary, and vanishes as

|x|\to\infty

. Five PT-symmetric potentials are studied: the Scarf-II potential

V_1(x)=iA_1\,{\rm sech}(x)\tanh(x)

, which decays exponentially for large

|x|

; the rational potentials

V_2(x)=iA_2\,x/(1+x^4)

and

V_3(x)=iA_3\,x/(1+|x|^3)

, which decay algebraically for large

|x|

; the step-function potential

V_4(x)=iA_4\,{\rm sgn}(x)θ(2.5-|x|)

, which has compact support; the regulated Coulomb potential

V_5(x)=iA_5\,x/(1+x^2)

, which decays slowly as

|x|\to\infty

and may be viewed as a long-range potential. The real parameters

A_n

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

+\infty

. The complex eigenvalues occur in discrete complex-conjugate pairs and for

V_n(x)

(

1\leq n\leq4

) the number of these pairs is finite and increases as the value of the strength parameter

A_n

increases. However, for

V_5(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.