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arXiv:quant-ph/0601188 · Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields

Equivalent Hermitian Hamiltonian for the non-Hermitian −x4 potential

Hugh F Jones, J. Mateo

Published Apr 5, 2006• 124 citations
arXivHTMLPDF

Abstract

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 $x$ is taken on a contour in the lower-half complex plane. It is then $PT$-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 $PT$ metric is not positive definite, but a dynamically-defined positive-definite metric can be defined, depending on an operator $Q$. Further, with the help of this operator an equivalent Hermitian Hamiltonian $h$ 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 ${m}^{2}{x}^{2}+ig{x}^{3}$. However, until now, the $\ensuremath{-}{x}^{4}$ potential has proved intractable. In the present paper we give explicit, closed form expressions for $Q$ and $h$, 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.

DOI
10.1103/physrevd.73.085002

Related papers

Equivalence of a complex PT -symmetric quartic Hamiltonian and a Hermitian quartic Hamiltonian with an anomaly

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In a recent paper Jones and Mateo used operator techniques to show that the non-Hermitian $\mathcal{P}\mathcal{T}$-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 $\mathcal{P}\mathcal{T}$-symmetric Hamiltonian. This anomaly in the Hermitian form of a $\mathcal{P}\mathcal{T}$-symmetric quartic Hamiltonian is unchanged if a harmonic term is introduced into $H$. 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.

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We present a generalization of the perturbative construction of the metric operator for non-Hermitian Hamiltonians with more than one perturbation parameter. We use this method to study the non-Hermitian scattering Hamiltonian H = p 2 /2 m + ζ − δ(x + α) + ζ + δ(x − α), where ζ ± and α are respectively complex and real parameters and δ(x) is the Dirac delta function. For regions in the space of coupling constants ζ ± where H is quasi-Hermitian and there are no complex bound states or spectral singularities, we construct a (positive-definite) metric operator η and the corresponding equivalent Hermitian Hamiltonian h . η turns out to be a (perturbatively) bounded operator for the cases where the imaginary part of the coupling constants have the opposite sign, ℑ(ζ + ) = −ℑ(ζ − ). This in particular contains the -symmetric case: ζ + = ζ* − . We also calculate the energy expectation values for certain Gaussian wave packets to study the nonlocal nature of h or equivalently the non-Hermitian nature of H . We show that these physical quantities are not directly sensitive to the presence of the -symmetry.

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6th International Workshop on Pseudo-Hermitian Hamiltonians in Quantum Physics

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Attempts to understand the quantum mechanics of non-Hermitian Hamiltonian systems can be traced back to the early days, one example being Heisenberg's endeavour to formulate a consistent model involving an indefinite metric. Over the years non-Hermitian Hamiltonians whose spectra were believed to be real have appeared from time to time in the literature, for instance in the study of strong interactions at high energies via Regge models, in condensed matter physics in the context of the XXZ-spin chain, in interacting boson models in nuclear physics, in integrable quantum field theories as Toda field theories with complex coupling constants, and also very recently in a field theoretical scenario in the quantization procedure of strings on an AdS 5 x S 5 background. Concrete experimental realizations of these types of systems in the form of optical lattices have been proposed in 2007. In the area of mathematical physics similar non-systematic results appeared sporadically over the years. However, intensive and more systematic investigation of these types of non- Hermitian Hamiltonians with real eigenvalue spectra only began about ten years ago, when the surprising discovery was made that a large class of one-particle systems perturbed by a simple non-Hermitian potential term possesses a real energy spectrum. Since then regular international workshops devoted to this theme have taken place. This special issue is centred around the 6th International Workshop on Pseudo-Hermitian Hamiltonians in Quantum Physics held in July 2007 at City University London. All the contributions contain significant new results or alternatively provide a survey of the state of the art of the subject or a critical assessment of the present understanding of the topic and a discussion of open problems. Original contributions from non-participants were also invited. Meanwhile many interesting results have been obtained and consensus has been reached on various central conceptual issues in the growing community of this subject. It is, for instance, well understood that the reality of the spectrum can be attributed either to the unbroken PT-symmetry of the entire system, that is, invariance of the Hamiltonian and the corresponding wavefunctions under a simultaneous parity transformation and time reversal, or more generally to its pseudo-Hermiticity . When the spectrum is real and discrete the Hamiltonian is actually quasi-Hermitian, with a positive-definite metric operator, and can in principle be related by a similarity transformation to an isospectral Hermitian counterpart. For all approaches well-defined procedures have been developed, which allow one to construct metric operators and therefore a consistent description of the underlying quantum mechanical observables. Even though the general principles have been laid out, it remains a challenge in most concrete cases to implement the entire procedure. Solvable models in this sense, some of which may be found in this issue, remain a rare exception. Nonetheless, despite this progress some important questions are still unanswered. For instance, according to the current understanding the non-Hermitian Hamiltonian does not uniquely define the physics of the system since a meaningful metric can no longer be associated with the system in a non-trivial and unambiguous manner. A fully consistent scattering theory has also not yet been formulated. Other issues remain controversial, such as the quantum brachistochrone problem, the problem of forming a mixture between a Hermitian and non-Hermitian system, the new phenomenological possibilities of forming a kind of worm-hole effect, etc. We would like to acknowledge the financial support of the London Mathematical Society, the Institute of Physics, the Doppler Institute in Prague and the School of Engineering and Mathematical Science of City University London. We hope this special issue will be useful to the newcomer as well as to the expert in the subject. Participants of the 6th International Workshop on Pseudo-Hermitian Hamiltonians in Quantum Physics.

Journal of Physics A Mathematical and Theoretical•Jun 3, 2008•13 citations
The potential V(x)=\ensuremath−x4V(x)=\ensuremath{-}{x}^{4}V(x)=\ensuremath−x4, 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\ensuremath{-}(ix{)}^{N}\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−x4\ensuremath{-}{x}^{4}\ensuremath−x4 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.
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.
We describe a method that allows for a practical application of the theory of pseudo-Hermitian operators to PT-symmetric systems defined on a complex contour. We apply this method to study the Hamiltonians H=p2+x2(ix)νH=p^2+x^2(ix)^νH=p2+x2(ix)ν with ν∈(−2,∞)ν\in(-2,\infty)ν∈(−2,∞) that are defined along the corresponding anti-Stokes lines. In particular, we reveal the intrinsic non-Hermiticity of HHH for the cases that ννν is an even integer, so that H=p2±x2+νH=p^2\pm x^{2+ν}H=p2±x2+ν, and give a proof of the discreteness of the spectrum of HHH for all ν∈(−2,∞)ν\in(-2,\infty)ν∈(−2,∞). Furthermore, we study the consequences of defining a square-well Hamiltonian on a wedge-shaped complex contour. This yields a PT-symmetric system with a finite number of real eigenvalues. We present a comprehensive analysis of this system within the framework of pseudo-Hermitian quantum mechanics. We also outline a direct pseudo-Hermitian treatment of PT-symmetric systems defined on a complex contour which clarifies the underlying mathematical structure of the formulation of PT-symmetric quantum mechanics based on the charge-conjugation operator. Our results provide a conclusive evidence that pseudo-Hermitian quantum mechanics provides a complete description of general PT-symmetric systems regardless of whether they are defined along the real line or a complex contour.
It is necessary to calculate the C\mathcal{C}C operator for the non-Hermitian PT\mathcal{P}\mathcal{T}PT-symmetric Hamiltonian $H=\frac{1}{2}{p}^{2}+\frac{1}{2}{\ensuremath{\mu}}^{2}{x}^{2}\ensuremath{-}\ensuremath{\lambda}{x}^{4}$ in order to demonstrate that HHH defines a consistent unitary theory of quantum mechanics. However, the C\mathcal{C}C operator cannot be obtained by using perturbative methods. Including a small imaginary cubic term gives the Hamiltonian $H=\frac{1}{2}{p}^{2}+\frac{1}{2}{\ensuremath{\mu}}^{2}{x}^{2}+ig{x}^{3}\ensuremath{-}\ensuremath{\lambda}{x}^{4}$, whose C\mathcal{C}C operator can be obtained perturbatively. In the semiclassical limit all terms in the perturbation series can be calculated in closed form and the perturbation series can be summed exactly. The result is a closed-form expression for C\mathcal{C}C having a nontrivial dependence on the dynamical variables xxx and ppp and on the parameter $\ensuremath{\lambda}$.
We present a generalization of the perturbative construction of the metric operator for non-Hermitian Hamiltonians with more than one perturbation parameter. We use this method to study the non-Hermitian scattering Hamiltonian H = p 2 /2 m + ζ − δ(x + α) + ζ + δ(x − α), where ζ ± and α are respectively complex and real parameters and δ(x) is the Dirac delta function. For regions in the space of coupling constants ζ ± where H is quasi-Hermitian and there are no complex bound states or spectral singularities, we construct a (positive-definite) metric operator η and the corresponding equivalent Hermitian Hamiltonian h . η turns out to be a (perturbatively) bounded operator for the cases where the imaginary part of the coupling constants have the opposite sign, ℑ(ζ + ) = −ℑ(ζ − ). This in particular contains the -symmetric case: ζ + = ζ* − . We also calculate the energy expectation values for certain Gaussian wave packets to study the nonlocal nature of h or equivalently the non-Hermitian nature of H . We show that these physical quantities are not directly sensitive to the presence of the -symmetry.
Attempts to understand the quantum mechanics of non-Hermitian Hamiltonian systems can be traced back to the early days, one example being Heisenberg's endeavour to formulate a consistent model involving an indefinite metric. Over the years non-Hermitian Hamiltonians whose spectra were believed to be real have appeared from time to time in the literature, for instance in the study of strong interactions at high energies via Regge models, in condensed matter physics in the context of the XXZ-spin chain, in interacting boson models in nuclear physics, in integrable quantum field theories as Toda field theories with complex coupling constants, and also very recently in a field theoretical scenario in the quantization procedure of strings on an AdS 5 x S 5 background. Concrete experimental realizations of these types of systems in the form of optical lattices have been proposed in 2007. In the area of mathematical physics similar non-systematic results appeared sporadically over the years. However, intensive and more systematic investigation of these types of non- Hermitian Hamiltonians with real eigenvalue spectra only began about ten years ago, when the surprising discovery was made that a large class of one-particle systems perturbed by a simple non-Hermitian potential term possesses a real energy spectrum. Since then regular international workshops devoted to this theme have taken place. This special issue is centred around the 6th International Workshop on Pseudo-Hermitian Hamiltonians in Quantum Physics held in July 2007 at City University London. All the contributions contain significant new results or alternatively provide a survey of the state of the art of the subject or a critical assessment of the present understanding of the topic and a discussion of open problems. Original contributions from non-participants were also invited. Meanwhile many interesting results have been obtained and consensus has been reached on various central conceptual issues in the growing community of this subject. It is, for instance, well understood that the reality of the spectrum can be attributed either to the unbroken PT-symmetry of the entire system, that is, invariance of the Hamiltonian and the corresponding wavefunctions under a simultaneous parity transformation and time reversal, or more generally to its pseudo-Hermiticity . When the spectrum is real and discrete the Hamiltonian is actually quasi-Hermitian, with a positive-definite metric operator, and can in principle be related by a similarity transformation to an isospectral Hermitian counterpart. For all approaches well-defined procedures have been developed, which allow one to construct metric operators and therefore a consistent description of the underlying quantum mechanical observables. Even though the general principles have been laid out, it remains a challenge in most concrete cases to implement the entire procedure. Solvable models in this sense, some of which may be found in this issue, remain a rare exception. Nonetheless, despite this progress some important questions are still unanswered. For instance, according to the current understanding the non-Hermitian Hamiltonian does not uniquely define the physics of the system since a meaningful metric can no longer be associated with the system in a non-trivial and unambiguous manner. A fully consistent scattering theory has also not yet been formulated. Other issues remain controversial, such as the quantum brachistochrone problem, the problem of forming a mixture between a Hermitian and non-Hermitian system, the new phenomenological possibilities of forming a kind of worm-hole effect, etc. We would like to acknowledge the financial support of the London Mathematical Society, the Institute of Physics, the Doppler Institute in Prague and the School of Engineering and Mathematical Science of City University London. We hope this special issue will be useful to the newcomer as well as to the expert in the subject. Participants of the 6th International Workshop on Pseudo-Hermitian Hamiltonians in Quantum Physics.