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arXiv:1908.03758 · Journal of Mathematical Physics

New families of non-parity-time-symmetric complex potentials with all-real spectra

Bijan Bagchi, Jianke Yang

Published Jun 1, 2020• 5 citations
arXivHTMLPDF

Abstract

New families of non-parity-time-symmetric complex potentials with all-real spectra are derived by the supersymmetry method and the pseudo-Hermiticity method. With the supersymmetry method, we find families of non-parity-time-symmetric complex partner potentials, which share the same spectrum as base potentials with known real spectra, such as the (complex) Wadati potentials. Different from previous supersymmetry derivations of potentials with real spectra, our derivation does not utilize discrete eigenmodes of base potentials. As a result, our partner potentials feature explicit analytical expressions, which contain free functions. With the pseudo-Hermiticity method, we derive a new class of non-parity-time-symmetric complex potentials with free functions and constants, whose eigenvalues appear as conjugate pairs. This eigenvalue symmetry forces the spectrum to be all-real for a wide range of choices of these functions and constants in the potential. Tuning these free functions and constants, phase transition can also be induced, where conjugate pairs of complex eigenvalues emerge in the spectrum.

DOI
10.1063/1.5124255

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New families of non-parity-time-symmetric complex potentials with all-real spectra are derived by the supersymmetry method and the pseudo-Hermiticity method. With the supersymmetry method, we find families of non-parity-time-symmetric complex partner potentials, which share the same spectrum as base potentials with known real spectra, such as the (complex) Wadati potentials. Different from previous supersymmetry derivations of potentials with real spectra, our derivation does not utilize discrete eigenmodes of base potentials. As a result, our partner potentials feature explicit analytical expressions, which contain free functions. With the pseudo-Hermiticity method, we derive a new class of non-parity-time-symmetric complex potentials with free functions and constants, whose eigenvalues appear as conjugate pairs. This eigenvalue symmetry forces the spectrum to be all-real for a wide range of choices of these functions and constants in the potential. Tuning these free functions and constants, phase transition can also be induced, where conjugate pairs of complex eigenvalues emerge in the spectrum.
A new form to construct complex superpotentials that produce real energy spectra in supersymmetric quantum mechanics is presented. This is based on the relation between the nonlinear Ermakov equation and a second order differential equation of the Schrödinger type. The superpotentials so constructed are characterized by the Ermakov parameters in such a way that they are always complex-valued and lead to non-Hermitian Hamiltonians with real spectra, whose eigenfunctions form a bi-orthogonal system. As applications we present new complex supersymmetric partners of the free particle that are P T -symmetric and can be either periodic or regular (of the Pöschl–Teller form). A new family of complex oscillators with real frequencies that have the energies of the harmonic oscillator plus an additional real eigenvalue is introduced.
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.
Changes of coordinates represent one of the most effective ways of deriving solvable potentials from ordinary differential equations for separate special functions. Here we relax the standard Hermiticity requirement and find an innovative construction which leads to unusual, complex potentials. Their energy spectrum is shown to stay real after a weakening of the Hermiticity of the Schrödinger equation to its mere invariance under the combined (parity) and (time-reversal) symmetry. This ultimately results in richer bound-state spectra. Some of our new exactly solvable potentials generalize the current textbook models. Details are given for constructions based on the hypergeometric and confluent hypergeometric special functions.
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=p2+x2(ix)ε of the harmonic oscillator Hamiltonian, where ε is a real parameter. The system exhibits two phases: When ε⩾0, the energy spectrum of H is real and positive as a consequence of 𝒫𝒯 symmetry. However, when −1<ε<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 ε=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=p2+x2N(ix)ε with N integer and ε>−N; each of these complex Hamiltonians exhibits a phase transition at ε=0. These 𝒫𝒯-symmetric theories may be viewed as analytic continuations of conventional theories from real to complex phase space.
We introduce and study a class of non-Hermitian Hamiltonians which have velocity dependent potentials. Since stability can not be advocated directly from the classical potential, we show that the energy spectra are real and bounded from below which proves the stability of the spectra of all members in the class. We find that the introduced class of non-Hermitian Hamiltonians do have a corresponding superpartner class of non-Hermitian Hamiltonians. We were able to introduce supercharges which in conjunction with the corresponding super Hamiltonians constitute a closed super algebra. Among the introduced Hamiltonians, we show that non-PT\mathcal{PT }PT-symmetric Hamiltonians can be transformed into their corresponding superpartner Hamiltonians via a specific canonical transformation while the PT\mathcal{PT }PT-symmetric ones failed to be mapped to their corresponding superpartner Hamiltonians via the same canonical transformation. Since canonical transformations preserve the spectrum, we conclude that non-PT\mathcal{PT }PT-symmetric Hamiltonians out of the introduced class of Hamiltonians have the same spectrum as the corresponding superpartner Hamiltonians and thus Susy is broken for such Hamiltonians. This kind of intertwining of PT\mathcal{PT }PT-symmetry and Supersymmetry is new as all the so far discussed cases concentrate on Hamiltonians of broken $\mathcal{PT }% −symmetrythathavebrokenSupersymmetrytoowhileweshowedthatSusycanbealsobrokenfornon−-symmetry that have broken Supersymmetry too while we showed that Susy can be also broken for non-−symmetrythathavebrokenSupersymmetrytoowhileweshowedthatSusycanbealsobrokenfornon−\mathcal{PT }$-symmetric and non-Hermitian Hamiltonians .
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.