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arXiv:1401.5937

An algebraically solvable PT-symmetric potential with broken symmetry

Ferreira, E. M., J. Sesma

Published Jan 23, 2014• 0 citations
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Abstract

The spectrum of a one-dimensional Hamiltonian with potential $V(x)=ix^2$ for negative $x$ and $V(x)=-ix^2$ for positive $x$ is analyzed. The Schrödinger equation is algebraically solvable and the eigenvalues are obtained as the zeros of an expression explicitly given in terms of Gamma functions. The spectrum consists of one real eigenvalue and an infinite set of pairs of complex conjugate eigenvalues.

DOI
10.48550/arxiv.1401.5937

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The spectrum of a one-dimensional Hamiltonian with potential V(x)=ix2V(x)=ix^2V(x)=ix2 for negative xxx and V(x)=−ix2V(x)=-ix^2V(x)=−ix2 for positive xxx is analyzed. The Schrödinger equation is algebraically solvable and the eigenvalues are obtained as the zeros of an expression explicitly given in terms of Gamma functions. The spectrum consists of one real eigenvalue and an infinite set of pairs of complex conjugate eigenvalues.
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.
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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.
We study a particular solvable potential and analyse the effect of 𝒫𝒯 symmetry on its bound state as well as scattering solutions. We determine the transmission and reflection coefficients for the 𝒫𝒯-symmetric case and also formulate the problem in terms of an SU (1,1) potential group, which allows unified treatment of the discrete and the continuous spectra in a natural way. We find that (bound and scattering) states of the 𝒫𝒯-symmetric problem supply a basis for the unitary irreducible representations of the SU (1,1) potential group, and this gives a straightforward group theoretical interpretation of the fact that the (complex) 𝒫𝒯-invariant potential has a real energy spectrum.