The energy spectrum of complex periodic potentials of the Kronig-Penney type

Hugh F Jones

Published Jan 1, 1999• 37 citations

Abstract

We consider a complex periodic PT-symmetric potential of the Kronig-Penney type, in order to elucidate the peculiar properties found by Bender et al. for potentials of the form $V=i(\sin x)^{2N+1}$, and in particular the absence of anti-periodic solutions. In this model we show explicitly why these solutions disappear as soon as $V^*(x)\neq V(x)$, and spell out the consequences for the form of the dispersion relation.

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The energy spectrum of complex periodic potentials of the Kronig-Penney type

Hugh F Jones

Published Jan 1, 1999• 37 citations

OpenAlex

Abstract

We consider a complex periodic PT-symmetric potential of the Kronig-Penney type, in order to elucidate the peculiar properties found by Bender et al. for potentials of the form

V=i(\sin x)^{2N+1}

, and in particular the absence of anti-periodic solutions. In this model we show explicitly why these solutions disappear as soon as

V^*(x)\neq V(x)

, and spell out the consequences for the form of the dispersion relation.

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arXiv:cond-mat/9909294Physics Letters ANov 1, 199952 citations

On the spectral properties of the Schrödinger operator with a periodic PT-symmetric potential

O. A. Veliev

In this paper, we investigate the spectrum and spectrality of the one-dimensional Schrödinger operator with a periodic PT-symmetric complex-valued potential.

arXiv:1609.05786International Journal of Geometric Methods in Modern PhysicsJan 10, 201720 citations

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American Journal of PhysicsDec 1, 199318 citations

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Recently, bootstrap methods from conformal field theory have been adapted for studying the energy spectrum of various quantum mechanical systems. In this paper, we consider the application of these methods in obtaining the spectrum from the Schr\"odinger equation with periodic potentials, paying particular attention to the Kronig-Penney model of a particle in a one-dimensional lattice. With an appropriate choice of operator basis involving position and momenta, we find that the bootstrap approach efficiently computes the band gaps of the energy spectrum but has trouble effectively constraining the minimum energy. We show how applying more complex constraints involving higher powers of momenta can potentially remedy such a problem. We also propose an approach for analytically constructing the dispersion relation associated with the Bloch momentum of the system.Recently, bootstrap methods from conformal field theory have been adapted for studying the energy spectrum of various quantum mechanical systems. In this paper, we consider the application of these methods in obtaining the spectrum from the Schr\"odinger equation with periodic potentials, paying particular attention to the Kronig-Penney model of a particle in a one-dimensional lattice. With an appropriate choice of operator basis involving position and momenta, we find that the bootstrap approach efficiently computes the band gaps of the energy spectrum but has trouble effectively constraining the minimum energy. We show how applying more complex constraints involving higher powers of momenta can potentially remedy such a problem. We also propose an approach for analytically constructing the dispersion relation associated with the Bloch momentum of the system.

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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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An algebraically solvable PT-symmetric potential with broken symmetry

Ferreira, E. M., J. Sesma

The spectrum of a one-dimensional Hamiltonian with potential $V(x)=ix^2$ for negative $x$ and $V(x)=-ix^2$ for positive $x$ is analyzed. The Schrödinger equation is algebraically solvable and the eigenvalues are obtained as the zeros of an expression explicitly given in terms of Gamma functions. The spectrum consists of one real eigenvalue and an infinite set of pairs of complex conjugate eigenvalues.The spectrum of a one-dimensional Hamiltonian with potential $V(x)=ix^2$ for negative $x$ and $V(x)=-ix^2$ for positive $x$ is analyzed. The Schrödinger equation is algebraically solvable and the eigenvalues are obtained as the zeros of an expression explicitly given in terms of Gamma functions. The spectrum consists of one real eigenvalue and an infinite set of pairs of complex conjugate eigenvalues.

arXiv:1401.5937Jan 23, 20140 citations

We consider a complex periodic PT-symmetric potential of the Kronig-Penney type, in order to elucidate the peculiar properties found by Bender et al. for potentials of the form

V=i(\sin x)^{2N+1}

, and in particular the absence of anti-periodic solutions. In this model we show explicitly why these solutions disappear as soon as

V^*(x)\neq V(x)

, and spell out the consequences for the form of the dispersion relation.

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.

The spectrum of a one-dimensional Hamiltonian with potential

V(x)=ix^2

for negative

x

and

V(x)=-ix^2

for positive

x

is analyzed. The Schrödinger equation is algebraically solvable and the eigenvalues are obtained as the zeros of an expression explicitly given in terms of Gamma functions. The spectrum consists of one real eigenvalue and an infinite set of pairs of complex conjugate eigenvalues.