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Physical Review A

Equivalent Hermitian Hamiltonians for some non-Hermitian Hamiltonians

Asiri Nanayakkara, Thilagarajah Mathanaranjan

Published Aug 7, 2012• 8 citations
Publisher

Abstract

Six years ago, by using operator techniques and path-integral methods, it was shown that the complex non-Hermitian $PT$-symmetric Hamiltonian ${p}^{2}\ensuremath{-}g{x}^{4}$ is equivalent to a conventional Hermitian Hamiltonian ${p}^{2}+4g{x}^{4}\ensuremath{-}2\ensuremath{\hbar}\sqrt{g}x$. Further it was revealed that the linear term in the Hermitian Hamiltonian is anomalous in the sense that it has no classical analog. In this paper we show that the complex non-Hermitian $\mathit{PT}$-symmetric Hamiltonian ${p}^{2}\ensuremath{-}g{x}^{4}+4i\ensuremath{\hbar}\sqrt{g}x$ and the conventional Hermitian Hamiltonian ${p}^{2}+4g{x}^{4}+6\ensuremath{\hbar}\sqrt{g}x$ have the same eigenspectra. In this case, the anomalous terms in both Hamiltonians are different from the previous one and vanish in the semiclassical limit. Further these equivalent Hamiltonians have zero-energy ground states. The exact ground-state wave functions and supersymmetric partner potentials are derived.

DOI
10.1103/physreva.86.022106

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Six years ago, by using operator techniques and path-integral methods, it was shown that the complex non-Hermitian PTPTPT-symmetric Hamiltonian p2\ensuremath−gx4{p}^{2}\ensuremath{-}g{x}^{4}p2\ensuremath−gx4 is equivalent to a conventional Hermitian Hamiltonian p2+4gx4\ensuremath−2\ensuremathℏgx{p}^{2}+4g{x}^{4}\ensuremath{-}2\ensuremath{\hbar}\sqrt{g}xp2+4gx4\ensuremath−2\ensuremathℏg​x. Further it was revealed that the linear term in the Hermitian Hamiltonian is anomalous in the sense that it has no classical analog. In this paper we show that the complex non-Hermitian PT\mathit{PT}PT-symmetric Hamiltonian p2\ensuremath−gx4+4i\ensuremathℏgx{p}^{2}\ensuremath{-}g{x}^{4}+4i\ensuremath{\hbar}\sqrt{g}xp2\ensuremath−gx4+4i\ensuremathℏg​x and the conventional Hermitian Hamiltonian p2+4gx4+6\ensuremathℏgx{p}^{2}+4g{x}^{4}+6\ensuremath{\hbar}\sqrt{g}xp2+4gx4+6\ensuremathℏg​x have the same eigenspectra. In this case, the anomalous terms in both Hamiltonians are different from the previous one and vanish in the semiclassical limit. Further these equivalent Hamiltonians have zero-energy ground states. The exact ground-state wave functions and supersymmetric partner potentials are derived.
In a recent paper Jones and Mateo used operator techniques to show that the non-Hermitian PT\mathcal{P}\mathcal{T}PT-symmetric wrong-sign quartic Hamiltonian $H=\frac{1}{2}{p}^{2}\ensuremath{-}g{x}^{4}$ has the same spectrum as the conventional Hermitian Hamiltonian $\stackrel{\texttildelow{}}{H}=\frac{1}{2}{p}^{2}+4g{x}^{4}\ensuremath{-}\sqrt{2g}x$. Here, this equivalence is demonstrated very simply by means of differential-equation techniques and, more importantly, by means of functional-integration techniques. It is shown that the linear term in the Hermitian Hamiltonian is anomalous; that is, this linear term has no classical analog. The anomaly arises because of the broken parity symmetry of the original non-Hermitian PT\mathcal{P}\mathcal{T}PT-symmetric Hamiltonian. This anomaly in the Hermitian form of a PT\mathcal{P}\mathcal{T}PT-symmetric quartic Hamiltonian is unchanged if a harmonic term is introduced into HHH. When there is a harmonic term, an immediate physical consequence of the anomaly is the appearance of bound states; if there were no anomaly term, there would be no bound states. Possible extensions of this work to $\ensuremath{-}{\ensuremath{\phi}}^{4}$ quantum field theory in higher-dimensional space-time are discussed.
It can be shown using operator techniques that the non-Hermitian PTPTPT-symmetric quantum mechanical Hamiltonian with a ``wrong-sign'' quartic potential $\ensuremath{-}g{x}^{4}$ is equivalent to a Hermitian Hamiltonian with a positive quartic potential together with a linear term. A na\"{\i}ve derivation of the same result in the path-integral approach misses this linear term. In a recent paper by Bender et al. it was pointed out that this term was in the nature of a parity anomaly and a more careful, discretized treatment of the path integral appeared to reproduce it successfully. However, on re-examination of this derivation we find that a yet more careful treatment is necessary, keeping terms that were ignored in that paper. An alternative, much simpler derivation is given using the additional potential that has been shown to appear whenever a change of variables to curvilinear coordinates is made in a functional integral.
The potential $V(x)=\ensuremath{-}{x}^{4}$, which is unbounded below on the real line, can give rise to a well-posed bound state problem when xxx is taken on a contour in the lower-half complex plane. It is then PTPTPT-symmetric rather than Hermitian. Nonetheless it has been shown numerically to have a real spectrum, and a proof of reality, involving the correspondence between ordinary differential equations and integrable systems, was subsequently constructed for the general class of potentials $\ensuremath{-}(ix{)}^{N}$. For such Hamiltonians the natural PTPTPT metric is not positive definite, but a dynamically-defined positive-definite metric can be defined, depending on an operator QQQ. Further, with the help of this operator an equivalent Hermitian Hamiltonian hhh can be constructed. This programme has been carried out exactly for a few soluble models, and the first few terms of a perturbative expansion have been found for the potential m2x2+igx3{m}^{2}{x}^{2}+ig{x}^{3}m2x2+igx3. However, until now, the $\ensuremath{-}{x}^{4}$ potential has proved intractable. In the present paper we give explicit, closed form expressions for QQQ and hhh, which are made possible by a particular parametrization of the contour in the complex plane on which the problem is defined. This constitutes an explicit proof of the reality of the spectrum. The resulting equivalent Hamiltonian has a potential with a positive quartic term together with a linear term.
PT\mathcal{PT}PT Symmetric Hamiltonian Model and Exactly Solvable Potentials
Searching for non-Hermitian (parity-time)PT\mathcal{PT}PT-symmetric Hamiltonians \cite{bender} with real spectra has been acquiring much interest for fourteen years. In this article, we have introduced a PT\mathcal{PT}PT symmetric non-Hermitian Hamiltonian model which is given as H^=ω(b^†b^+12)+α(b^2−(b^†)2)\hat{\mathcal{H}}=ω(\hat{b}^\dagger\hat{b}+\frac{1}{2})+ α(\hat{b}^{2}-(\hat{b}^\dagger)^{2})H^=ω(b^†b^+21​)+α(b^2−(b^†)2) where ωωω and ααα are real constants, b^\hat{b}b^ and b†^\hat{b^\dagger}b†^ are first order differential operators. Moreover, Pseudo-Hermiticity that is a generalization of PT\mathcal{PT}PT symmetry has been attracting a growing interest \cite{mos}. Because the Hamiltonian H\mathcal{H}H is pseudo-Hermitian, we have obtained the Hermitian equivalent of H\mathcal{H}H which is in Sturm- Liouville form leads to exactly solvable potential models which are effective screened potential and hyperbolic Rosen-Morse II potential. H\mathcal{H}H is called pseudo-Hermitian, if there exists a Hermitian and invertible operator ηηη satisfying H†=ηHη−1\mathcal{H^\dagger}=η\mathcal{H} η^{-1}H†=ηHη−1. For the Hermitian Hamiltonian hhh, one can write h=ρHρ−1h=ρ\mathcal{H} ρ^{-1}h=ρHρ−1 where ρ=ηρ=\sqrtηρ=η​ is unitary. Using this ρρρ we have obtained a physical Hamiltonian hhh for each case. Then, the Schrödinger equation is solved exactly using Shape Invariance method of Supersymmetric Quantum Mechanics \cite{susy1}. Mapping function ρρρ is obtained for each potential case.
This thesis is centred around the role of non-Hermitian Hamiltonians in Physics both at the quantum and classical levels. In our investigations of two-level models we demonstrate [1] the phenomenon of fast transitions developed in the PT -symmetric quantum brachistochrone problem may in fact be attributed to the non-Hermiticity of evolution operator used, rather than to its invariance under PT operation. Transition probabilities are calculated for Hamiltonians which explicitly violate PT -symmetry. When it comes to Hilbert spaces of infinite dimension, starting with non-Hermitian Hamiltonians expressed as linear and quadratic combinations of the generators of the su(1; 1) Lie algebra, we construct [2] Hermitian partners in the same similarity class. Alongside, metrics with respect to which the original Hamiltonians are Hermitian are also constructed, allowing to assign meaning to a large class of non-Hermitian Hamiltonians possessing real spectra. The finding of exact results to establish the physical acceptability of other non-Hermitian models may be pursued by other means, especially if the system of interest cannot be expressed in terms of Lie algebraic elements. We also employ [3] a representation of the canonical commutation relations for position and momentum operators in terms of real-valued functions and a noncommutative product rule of differential form. Besides exact solutions, we also compute in a perturbative fashion metrics and isospectral partners for systems of physical interest. Classically, our efforts were concentrated on integrable models presenting PT - symmetry. Because the latter can also establish the reality of energies in classical systems described by Hamiltonian functions, we search for new families of nonlinear differential equations for which the presence of hidden symmetries allows one to assemble exact solutions. We use [4] the Painleve test to check whether deformations of integrable systems preserve integrability. Moreover we compare [5] integrable deformed models, which are thus likely to possess soliton solutions, to a broader class of systems presenting compacton solutions. Finally we study [6] the pole structure of certain real valued nonlinear integrable systems and establish that they behave as interacting particles whose motion can be extended to the complex plane in a PT -symmetric way.
This thesis is centred around the role of non-Hermitian Hamiltonians in Physics both at the quantum and classical levels. In our investigations of two-level models we demonstrate [1] the phenomenon of fast transitions developed in the PT -symmetric quantum brachistochrone problem may in fact be attributed to the non-Hermiticity of evolution operator used, rather than to its invariance under PT operation. Transition probabilities are calculated for Hamiltonians which explicitly violate PT -symmetry. When it comes to Hilbert spaces of infinite dimension, starting with non-Hermitian Hamiltonians expressed as linear and quadratic combinations of the generators of the su(1; 1) Lie algebra, we construct [2] Hermitian partners in the same similarity class. Alongside, metrics with respect to which the original Hamiltonians are Hermitian are also constructed, allowing to assign meaning to a large class of non-Hermitian Hamiltonians possessing real spectra. The finding of exact results to establish the physical acceptability of other non-Hermitian models may be pursued by other means, especially if the system of interest cannot be expressed in terms of Lie algebraic elements. We also employ [3] a representation of the canonical commutation relations for position and momentum operators in terms of real-valued functions and a noncommutative product rule of differential form. Besides exact solutions, we also compute in a perturbative fashion metrics and isospectral partners for systems of physical interest. Classically, our efforts were concentrated on integrable models presenting PT - symmetry. Because the latter can also establish the reality of energies in classical systems described by Hamiltonian functions, we search for new families of nonlinear differential equations for which the presence of hidden symmetries allows one to assemble exact solutions. We use [4] the Painleve test to check whether deformations of integrable systems preserve integrability. Moreover we compare [5] integrable deformed models, which are thus likely to possess soliton solutions, to a broader class of systems presenting compacton solutions. Finally we study [6] the pole structure of certain real valued nonlinear integrable systems and establish that they behave as interacting particles whose motion can be extended to the complex plane in a PT -symmetric way.