TY - JOUR

T1 - PT-symmetric potentials having continuous spectra

AU - Wen, Zichao

AU - Bender, Carl M.

N1 - Publisher Copyright:
© 2020 IOP Publishing Ltd.

PY - 2020/9/18

Y1 - 2020/9/18

N2 - One-dimensional Pt-symmetric quantum-mechanical Hamiltonians having continuous spectra are studied. The Hamiltonians considered have the form H = p 2 + V(x), where-∞ < x < ∞. The potential V(x) is odd in x, pure imaginary, and vanishes as |x| → ∞. Five PT-symmetric potentials are studied: the Scarf-II potential V 1(x) = iA 1 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 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| → ∞ and thus 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, in general, the spectra of these Hamiltonians 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 V n (x) (1 n 4) 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 infinite sequence of discrete eigenvalues with a limit point at the origin. While this sequence is complex, it resembles the Balmer series for the hydrogen atom because it has inverse-square convergence.

AB - One-dimensional Pt-symmetric quantum-mechanical Hamiltonians having continuous spectra are studied. The Hamiltonians considered have the form H = p 2 + V(x), where-∞ < x < ∞. The potential V(x) is odd in x, pure imaginary, and vanishes as |x| → ∞. Five PT-symmetric potentials are studied: the Scarf-II potential V 1(x) = iA 1 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 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| → ∞ and thus 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, in general, the spectra of these Hamiltonians 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 V n (x) (1 n 4) 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 infinite sequence of discrete eigenvalues with a limit point at the origin. While this sequence is complex, it resembles the Balmer series for the hydrogen atom because it has inverse-square convergence.

KW - Continuous spectra

KW - Long-range potentials

KW - NonHermitian

KW - PT symmetry

UR - http://www.scopus.com/inward/record.url?scp=85090422605&partnerID=8YFLogxK

U2 - 10.1088/1751-8121/aba468

DO - 10.1088/1751-8121/aba468

M3 - Article

AN - SCOPUS:85090422605

SN - 1751-8113

VL - 53

JO - Journal of Physics A: Mathematical and Theoretical

JF - Journal of Physics A: Mathematical and Theoretical

IS - 37

M1 - 375302

ER -