arXiv:1507.03644

Three PT-symmetric Hamiltonians with completely different spectra

Francisco Marcelo Fernández, Javier Garcia

Published Jul 13, 2015• 0 citations

Abstract

We discuss three Hamiltonians, each with a central-field part $H_{0}$ and a PT-symmetric perturbation $igz$. When $H_{0}$ is the isotropic Harmonic oscillator the spectrum is real for all $g$ because $H$ is isospectral to $H_{0}+g^{2}/2$. When $H_{0}$ is the Hydrogen atom then infinitely many eigenvalues are complex for all $g$. If the potential in $H_{0}$ is linear in the radial variable $r$ then the spectrum of $H$ exhibits real eigenvalues for $0

DOI: 10.48550/arxiv.1507.03644

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arXiv:1507.03644

Three PT-symmetric Hamiltonians with completely different spectra

Francisco Marcelo Fernández, Javier Garcia

Published Jul 13, 2015• 0 citations

arXiv HTML PDF

Abstract

We discuss three Hamiltonians, each with a central-field part

H_{0}

and a PT-symmetric perturbation

igz

. When

H_{0}

is the isotropic Harmonic oscillator the spectrum is real for all

g

because

H

is isospectral to

H_{0}+g^{2}/2

. When

H_{0}

is the Hydrogen atom then infinitely many eigenvalues are complex for all

g

. If the potential in

H_{0}

is linear in the radial variable

r

then the spectrum of

H

exhibits real eigenvalues for $0

DOI: 10.48550/arxiv.1507.03644

Related papers

Fundamental Properties of Eigenstate Wave Functions of PT-Symmetric Anharmonic Oscillators

Jack H. Noble, Ulrich D. Jentschura

The PT-symmetric cubic oscillator with Hamiltonian H_3 = - (1/2) d_x^2 + (1/2) x^2 + i G x^3 is a paradigmatic example of a pseudo-Hermitian, or PT-symmetric Hamiltonian with a purely real spectrum when endowed with L^2(R) boundary conditions. Eigenfunctions of the stationary Schr\{o}dinger equation H_3 psi_n(x) = E_n psi_n(x) are manifestly complex, while the energy eigenvalues E_n are real and positive. Although H_3 does not commute with the parity operator, we find that for a natural choice of the global complex phase of the eigenstate wave function, the real and imaginary parts of the eigenfunctions [i.e., Re psi_n(x) and Im psi_n(x)] are eigenstates of parity. Both the real and imaginary parts of the eigenfunctions are found to have an infinite number of zeros, even for the ground state and even for infinitesimal G, but the real and imaginary parts of psi_n(x) never vanish simultaneously when G>0. Furthermore, we find that the eigenfunctions are concentrated in an allowed region where the energy E_n is larger than the complex modulus of the complex potential. PT-symmetric Hamiltonians constitute natural generalizations of Hermitian Hamiltonians as time-evolution operators. Our results suggest that PT-symmetric (pseudo-Hermitian) time evolution can naturally be interpreted as time evolution in a situation where manifestly complex gain and loss terms mutually compensate and lead to the real energy eigenvalues.The PT-symmetric cubic oscillator with Hamiltonian H_3 = - (1/2) d_x^2 + (1/2) x^2 + i G x^3 is a paradigmatic example of a pseudo-Hermitian, or PT-symmetric Hamiltonian with a purely real spectrum when endowed with L^2(R) boundary conditions. Eigenfunctions of the stationary Schr\{o}dinger equation H_3 psi_n(x) = E_n psi_n(x) are manifestly complex, while the energy eigenvalues E_n are real and positive. Although H_3 does not commute with the parity operator, we find that for a natural choice of the global complex phase of the eigenstate wave function, the real and imaginary parts of the eigenfunctions [i.e., Re psi_n(x) and Im psi_n(x)] are eigenstates of parity. Both the real and imaginary parts of the eigenfunctions are found to have an infinite number of zeros, even for the ground state and even for infinitesimal G, but the real and imaginary parts of psi_n(x) never vanish simultaneously when G>0. Furthermore, we find that the eigenfunctions are concentrated in an allowed region where the energy E_n is larger than the complex modulus of the complex potential. PT-symmetric Hamiltonians constitute natural generalizations of Hermitian Hamiltonians as time-evolution operators. Our results suggest that PT-symmetric (pseudo-Hermitian) time evolution can naturally be interpreted as time evolution in a situation where manifestly complex gain and loss terms mutually compensate and lead to the real energy eigenvalues.

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Carl M. Bender, Daniel W. Hook, Lawrence R Mead

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We discuss three Hamiltonians, each with a central-field part

H_{0}

and a PT-symmetric perturbation

igz

. When

H_{0}

is the isotropic Harmonic oscillator the spectrum is real for all

g

because

H

is isospectral to

H_{0}+g^{2}/2

. When

H_{0}

is the Hydrogen atom then infinitely many eigenvalues are complex for all

g

. If the potential in

H_{0}

is linear in the radial variable

r

then the spectrum of

H

exhibits real eigenvalues for $0

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.