The spectrum of the Hermitian Hamiltonian ${1\\over2}p^2+{1\\over2}m^2x^2+gx^4$ ( $g>0$ ), which describes the quantum anharmonic oscillator, is real and positive. The non-Hermitian quantum-mechanical Hamiltonian $H={1\\over2}p^2+{1 \\over2}m^2x^2-gx^4$ , where the coupling constant $g$ is real and positive, is ${\\cal PT}$ -symmetric. As a consequence, the spectrum of $H$ is known to be real and positive as well. Here, it is shown that there is a significant difference between these two theories: When $g$ is sufficiently small, the latter Hamiltonian exhibits a two-particle bound state while the former does not. The bound state persists in the corresponding non-Hermitian ${\\cal PT}$ -symmetric $-g\\phi^4$ quantum field theory for all dimensions $0\\leq D<3$ but is not present in the conventional Hermitian $g\\phi^4$ field theory.The spectrum of the Hermitian Hamiltonian ${1\\over2}p^2+{1\\over2}m^2x^2+gx^4$ ($g>0$), which describes the quantum anharmonic oscillator, is real and positive. The non-Hermitian quantum-mechanical Hamiltonian $H={1\\over2}p^2+{1 \\over2}m^2x^2-gx^4$, where the coupling constant $g$ is real and positive, is ${\\cal PT}$-symmetric. As a consequence, the spectrum of $H$ is known to be real and positive as well. Here, it is shown that there is a significant difference between these two theories: When $g$ is sufficiently small, the latter Hamiltonian exhibits a two-particle bound state while the former does not. The bound state persists in the corresponding non-Hermitian ${\\cal PT}$-symmetric $-g\\phi^4$ quantum field theory for all dimensions $0\\leq D<3$ but is not present in the conventional Hermitian $g\\phi^4$ field theory.

Jan 1, 200123 citations

Non-Hermitian quantum mechanics: the case of bound state scattering theory

A. Matzkin

Excited bound states are often understood within scattering-based theories as resulting from the collision of a particle on a target via a short-range potential. We show that the resulting formalism is non-Hermitian and describe the Hilbert spaces and metric operator relevant to a correct formulation of such theories. The structure and tools employed are the same that have been introduced in current works dealing with PT-symmetric and quasi-Hermitian problems. The relevance of the non-Hermitian formulation to practical computations is assessed by introducing a non-Hermiticity index. We give a numerical example involving scattering by a short-range potential in a Coulomb field for which it is seen that even for a small but non-negligible non-Hermiticity index the non-Hermitian character of the problem must be taken into account. The computation of physical quantities in the relevant Hilbert spaces is also discussed.Excited bound states are often understood within scattering-based theories as resulting from the collision of a particle on a target via a short-range potential. We show that the resulting formalism is non-Hermitian and describe the Hilbert spaces and metric operator relevant to a correct formulation of such theories. The structure and tools employed are the same that have been introduced in current works dealing with PT-symmetric and quasi-Hermitian problems. The relevance of the non-Hermitian formulation to practical computations is assessed by introducing a non-Hermiticity index. We give a numerical example involving scattering by a short-range potential in a Coulomb field for which it is seen that even for a small but non-negligible non-Hermiticity index the non-Hermitian character of the problem must be taken into account. The computation of physical quantities in the relevant Hilbert spaces is also discussed.

arXiv:quant-ph/0603238Journal of Physics A Mathematical and GeneralAug 9, 200618 citations

Nonrelativistic Quantum Field Theory

Walter Greiner, J. Reinhardt

Springer eBooksJan 1, 19961 citation

Non-Hermitian Hamiltonians in Quantum Physics

Fabio Bagarello, Roberto Passante, Camillo Trapani

Springer proceedings in physicsJan 1, 2016111 citations

The spectrum of the Hermitian Hamiltonian

{1\\over2}p^2+{1\\over2}m^2x^2+gx^4

(

g>0

), which describes the quantum anharmonic oscillator, is real and positive. The non-Hermitian quantum-mechanical Hamiltonian

H={1\\over2}p^2+{1 \\over2}m^2x^2-gx^4

, where the coupling constant

g

is real and positive, is

{\\cal PT}

-symmetric. As a consequence, the spectrum of

H

is known to be real and positive as well. Here, it is shown that there is a significant difference between these two theories: When

g

is sufficiently small, the latter Hamiltonian exhibits a two-particle bound state while the former does not. The bound state persists in the corresponding non-Hermitian

{\\cal PT}

-symmetric

-g\\phi^4

quantum field theory for all dimensions

0\\leq D<3

but is not present in the conventional Hermitian

g\\phi^4

field theory.