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arXiv:1203.6590 · Journal of Physics A Mathematical and Theoretical

Negative-energy $\mathcal {P}\mathcal {T}$-symmetric Hamiltonians

Carl M. Bender, Daniel W. Hook, S. P. Klevansky

Published Oct 23, 2012• 9 citations
arXivHTMLPDF

Abstract

The non-Hermitian PT-symmetric quantum-mechanical Hamiltonian $H=p^2+x^2(ix)^ε$ has real, positive, and discrete eigenvalues for all $ε\geq 0$. These eigenvalues are analytic continuations of the harmonic-oscillator eigenvalues $E_n=2n+1$ (n=0, 1, 2, 3, ...) at $ε=0$. However, the harmonic oscillator also has negative eigenvalues $E_n=-2n-1$ (n=0, 1, 2, 3, ...), and one may ask whether it is equally possible to continue analytically from these eigenvalues. It is shown in this paper that for appropriate PT-symmetric boundary conditions the Hamiltonian $H=p^2+x^2(ix)^ε$ also has real and {\it negative} discrete eigenvalues. The negative eigenvalues fall into classes labeled by the integer N (N=1, 2, 3, ...). For the Nth class of eigenvalues, $ε$ lies in the range $(4N-6)/3

DOI
10.1088/1751-8113/45/44/444003

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Negative-energy PT\mathcal {P}\mathcal {T}PT-symmetric Hamiltonians
The non-Hermitian PT-symmetric quantum-mechanical Hamiltonian H=p2+x2(ix)εH=p^2+x^2(ix)^εH=p2+x2(ix)ε has real, positive, and discrete eigenvalues for all ε≥0ε\geq 0ε≥0. These eigenvalues are analytic continuations of the harmonic-oscillator eigenvalues En=2n+1E_n=2n+1En​=2n+1 (n=0, 1, 2, 3, ...) at ε=0ε=0ε=0. However, the harmonic oscillator also has negative eigenvalues En=−2n−1E_n=-2n-1En​=−2n−1 (n=0, 1, 2, 3, ...), and one may ask whether it is equally possible to continue analytically from these eigenvalues. It is shown in this paper that for appropriate PT-symmetric boundary conditions the Hamiltonian H=p2+x2(ix)εH=p^2+x^2(ix)^εH=p2+x2(ix)ε also has real and {\it negative} discrete eigenvalues. The negative eigenvalues fall into classes labeled by the integer N (N=1, 2, 3, ...). For the Nth class of eigenvalues, εεε lies in the range $(4N-6)/3
This paper proposes to broaden the canonical formulation of quantum mechanics.Ordinarily, one imposes the condition H † = H on the Hamiltonian, where † represents the mathematical operation of complex conjugation and matrix transposition.This conventional Hermiticity condition is sufficient to ensure that the Hamiltonian H has a real spectrum.However, replacing this mathematical condition by the weaker and more physical requirement H ‡ = H, where ‡ represents combined parity reflection and time reversal PT , one obtains new classes of complex Hamiltonians whose spectra are still real and positive.This generalization of Hermiticity is investigated using a complex deformation H = p 2 + x 2 (ix) ǫ of the harmonic oscillator Hamiltonian, where ǫ is a real parameter.The system exhibits two phases: When ǫ ≥ 0, the energy spectrum of H is real and positive as a consequence of PT symmetry.However, when -1 < ǫ < 0, the spectrum contains an infinite number of complex eigenvalues and a finite number of real, positive eigenvalues because PT symmetry is spontaneously broken.The phase transition that occurs at ǫ = 0 manifests itself in both the quantum-mechanical system and the underlying classical system.Similar qualitative features are exhibited by complex deformations of other standard real Hamiltonians H = p 2 + x 2N (ix) ǫ with N integer and ǫ > -N ; each of these complex Hamiltonians exhibits a phase transition at ǫ = 0.These PT -symmetric theories may be viewed as analytic continuations of conventional theories from real to complex phase space.
Most quantum physicists believe that a quantum-mechanical Hamiltonian must be Dirac Hermitian (invariant under matrix transposition and complex conjugation) to be sure that the energy eigenvalues are real and that time evolution is unitary. However, the non-Dirac-hermitian Hamiltonian H = p2+ix3 has a real positive discrete spectrum and generates unitary time evolution and defines a fully consistent and physical quantum theory. Evidently, Dirac Hermiticity is too restrictive. While H = p2+ix3 is not Dirac Hermitian, it is PT symmetric (invariant under combined space reflection P and time reversal T). Another PT-symmetric Hamiltonian whose energy levels are real, positive and discrete is H = p2−x4, which contains an upside-down potential. The quantum mechanics defined by a PT-symmetric Hamiltonian is a complex generalization of ordinary quantum mechanics. When quantum mechanics and quantum field theory are extended into the complex domain, new kinds of theories having strange and remarkable properties emerge. In the past two years some of these properties have been verified in laboratory experiments. Here, we first discuss PT-symmetric Hamiltonians at a simple intuitive level and explain why the energy levels of such Hamiltonians may be real, positive, and discrete. Second, we describe a recent experiment in which the PT phase transition was observed. Third, we briefly mention that PT-symmetric theories can be useful at a fundamental level. While the double-scaling limit of an O(N)-symmetric gφ4 quantum field theory appears to be inconsistent because the critical value of g is negative, this limit is in fact not inconsistent because the critical theory is PT symmetric.
In this thesis we aim to show that the Schrödinger equation, which is a boundary eigenvalue problem, can have a discrete and real energy spectrum (eigenvalues) even when the Hamiltonian is non-Hermitian.After a brief introduction into non-Hermiticity, we will focus on solving the Schrödinger equation with a special class of non-Hermitian Hamiltonians, namely PTsymmetric Hamiltonians.PT -symmetric Hamiltonians have been discussed by various authors [1, 2, 3, 4, 5] with some of them focusing specifically on obtaining the real and discrete energy spectrum.Various methods for solving this problematic Schrödinger equation will be considered.After starting with perturbation theory, we will move on to numerical methods.Three different categories of methods will be discussed.First there is the shooting method based on a Runge-Kutta solver.Next, we investigate various implementations of the spectral method.Finally, we will look at the Riccati-Padé method, which is a numerical implemented analytical method.PT -symmetric potentials need to be solved along a contour in the complex plane.We will propose modifications to the numerical methods to handle this.After solving the widely documented PT -symmetric Hamiltonian H = p 2 -(ix) N with these methods, we give a discussion and comparison of the obtained results.Finally, we solve another PT -symmetric potential, illustrating the use of paths in the complex plane to obtain a real and discrete spectrum and their influence on the results.
One-dimensional PT-symmetric quantum-mechanical Hamiltonians having continuous spectra are studied. The Hamiltonians considered have the form H=p2+V(x)H=p^2+V(x)H=p2+V(x), where V(x)V(x)V(x) is odd in xxx, pure imaginary, and vanishes as ∣x∣→∞|x|\to\infty∣x∣→∞. Five PT-symmetric potentials are studied: the Scarf-II potential V1(x)=iA1 sech(x)tanh⁡(x)V_1(x)=iA_1\,{\rm sech}(x)\tanh(x)V1​(x)=iA1​sech(x)tanh(x), which decays exponentially for large ∣x∣|x|∣x∣; the rational potentials V2(x)=iA2 x/(1+x4)V_2(x)=iA_2\,x/(1+x^4)V2​(x)=iA2​x/(1+x4) and V3(x)=iA3 x/(1+∣x∣3)V_3(x)=iA_3\,x/(1+|x|^3)V3​(x)=iA3​x/(1+∣x∣3), which decay algebraically for large ∣x∣|x|∣x∣; the step-function potential V4(x)=iA4 sgn(x)θ(2.5−∣x∣)V_4(x)=iA_4\,{\rm sgn}(x)θ(2.5-|x|)V4​(x)=iA4​sgn(x)θ(2.5−∣x∣), which has compact support; the regulated Coulomb potential V5(x)=iA5 x/(1+x2)V_5(x)=iA_5\,x/(1+x^2)V5​(x)=iA5​x/(1+x2), which decays slowly as ∣x∣→∞|x|\to\infty∣x∣→∞ and may be viewed as a long-range potential. The real parameters AnA_nAn​ 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 000 to +∞+\infty+∞. The complex eigenvalues occur in discrete complex-conjugate pairs and for Vn(x)V_n(x)Vn​(x) (1≤n≤41\leq n\leq41≤n≤4) the number of these pairs is finite and increases as the value of the strength parameter AnA_nAn​ increases. However, for V5(x)V_5(x)V5​(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 average quantum physicist on the street would say that a quantum-mechanical Hamiltonian must be Dirac Hermitian (invariant under combined matrix transposition and complex conjugation) in order to guarantee that the energy eigenvalues are real and that time evolution is unitary. However, the Hamiltonian H = p 2 + ix 3 , which is obviously not Dirac Hermitian, has a positive real discrete spectrum and generates unitary time evolution, and thus it defines a fully consistent and physical quantum theory. Evidently, the axiom of Dirac Hermiticity is too restrictive. While H = p 2 + ix 3 is not Dirac Hermitian, it is PT symmetric; that is, invariant under combined parity P (space reflection) and time reversal T . The quantum mechanics defined by a PT -symmetric Hamiltonian is a complex generalization of ordinary quantum mechanics. When quantum mechanics is extended into the complex domain, new kinds of theories having strange and remarkable properties emerge. In the past few years, some of these properties have been verified in laboratory experiments. A particularly interesting PT-symmetric Hamiltonian is H = p 2 − x 4 , which contains an upside-down potential. This potential is discussed in detail, and it is explained in intuitive as well as in rigorous terms why the energy levels of this potential are real, positive, and discrete. Applications of PT -symmetry in quantum field theory are also discussed.