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arXiv:1810.00578 · Physical Review A

Superoscillatory PT -symmetric potentials

Yaniv Eliezer, Alon Bahabad, Boris A. Malomed

Published Oct 15, 2018• 4 citations
arXivHTMLPDF

Abstract

We introduce the one-dimensional $\mathcal{PT}$-symmetric Schr\"odinger equation, with complex potentials in the form of the canonical superoscillatory and suboscillatory functions known in quantum mechanics and optics. While the suboscillatorylike potential always generates an entirely real eigenvalue spectrum, its counterpart based on the superoscillatory wave function gives rise to an intricate pattern of $\mathcal{PT}$-symmetry-breaking transitions, controlled by the parameters of the superoscillatory function. One scenario of the transitions proceeds smoothly via a set of threshold values, while another one exhibits a sudden jump to the broken $\mathcal{PT}$ symmetry. Another noteworthy finding is the possibility of restoration of the $\mathcal{PT}$ symmetry, following its original loss, in the course of the variation of the parameters.

DOI
10.1103/physreva.98.043830

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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.
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 consider PT-symmetric quantum mechanical potentials, having an underlying supersymmetry (SUSY). The eigenspectra of both the broken and unbroken phases of PT-symmetry are analytically obtained through SUSY and shape invariance (SI). The SI is characterized by real and imaginary parametric shifting for unbroken and broken phases of PT, respectively. The use of SUSY and PT is shown to yield several complex potential systems, originating from Coulomb, Poschl-Teller, and other solvable problems. In the PT-symmetric phase, spontaneous breaking of SUSY is observed in some parametric domains, which leads to non-trivial shape invariances to obtain the energy spectra. A conserved non-local correlation in PT-symmetric problems is shown to explain observed transmission and reflection behavior in optical systems, possessing PT-symmetry. Interestingly, the stationary states with real energy emerge in the PT-symmetric phase. The correlation also plays a crucial role in the broken PT case, having the complex conjugate energy eigenvalues and the corresponding states related.
A major mathematical problem in PT-symmetric quantum mechanics is to determine whether or not the spectrum of any given non self-adjoint but PT-symmetric Schroedinger operator is real. In this case the PT-symmetry is called "proper". Clearly in this connection an equally important issue is the spontaneous violation of the PT-symmetry which might occur in a PT-symmetric operator family. The spontaneous violation of the PT-symmetry is defined as the transition from real values of the spectrum to complex ones at the variation of the parameter labeling the family. Its occurrence is referred to also as the PTPTPT-symmetric phase transition. \nThis chapter intends to present a review of recent results concerning these two mathematical points, within the standard notions of spectral theory for Hilbert space operators. The main technical instrument is represented by perturbation theory.
We study complex PT-symmetric potentials, with real eigenvalues, corresponding to a complex coordinate shift (x+ic2) of a real even potential. In this case,the rules to achieve a coherent quantum mechanics are known. They allow the calculation of observables, which are found to be independent on c. This result is illustratedby few analytical or semi analytical examples. On the other hand, trying to test thisproperty numerically faces problems linked to the difficulty of finding the proper solutions of the Schrodinger equation. In particular, the large distance behaviour of the ¨wave functions generates instabilities. As an example, we have studied the (x +ic2)4potential.