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

Nonrelativistic Quantum Field Theory

Walter Greiner, J. Reinhardt

Springer eBooksJan 1, 19961 citation

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

A General Quantum Speed Limit for Non-Hermitian Systems

Zhanxi Wang, Xiaozhe Hao, X. X. Yi

The quantum speed limit (QSL) refers to the maximum speed of a quantum system to evolve from an initial state to its orthogonal states. The bound on the QSL for Hermitian systems, for example the Mandelstam-Tamm (MT) and Margolus-Levitin (ML) as well as Sun-Zheng(SZ) bound, was studied respectively from the perspectives of average value and variance of the system Hamiltonian as well as the geometry of the system. While the compactness of the MT-type, ML-type and SZ-type bounds has been examined well for Hermitian systems, a compact QSL for non-Hermitian systems has not been well studied. In this work, based on the biorthogonal basis theory we derive two distinct and tighter bounds on the QSL for non-Hermitian systems, which correspond to the MT and ML bounds for Hermitian systems. We show that the shortest evolution time corresponding to the two bounds of the non-Hermitian system can be attained by certain initial states, showing the compactness and tightness of our bounds. These initial states dubbed fastest initial states(FIS) are different from that in Hermitian systems. A bound close to QSL for non-FIS is presented and comparison of our bound with others in literature is performed. To illustrate our results, we present a minimal non-Hermitian system to show QSL, and the condition for the shortest evolution time is derived analytically using the present theory.The quantum speed limit (QSL) refers to the maximum speed of a quantum system to evolve from an initial state to its orthogonal states. The bound on the QSL for Hermitian systems, for example the Mandelstam-Tamm (MT) and Margolus-Levitin (ML) as well as Sun-Zheng(SZ) bound, was studied respectively from the perspectives of average value and variance of the system Hamiltonian as well as the geometry of the system. While the compactness of the MT-type, ML-type and SZ-type bounds has been examined well for Hermitian systems, a compact QSL for non-Hermitian systems has not been well studied. In this work, based on the biorthogonal basis theory we derive two distinct and tighter bounds on the QSL for non-Hermitian systems, which correspond to the MT and ML bounds for Hermitian systems. We show that the shortest evolution time corresponding to the two bounds of the non-Hermitian system can be attained by certain initial states, showing the compactness and tightness of our bounds. These initial states dubbed fastest initial states(FIS) are different from that in Hermitian systems. A bound close to QSL for non-FIS is presented and comparison of our bound with others in literature is performed. To illustrate our results, we present a minimal non-Hermitian system to show QSL, and the condition for the shortest evolution time is derived analytically using the present theory.

quant-pharXiv:2605.23250May 25, 20260 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.