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Iranian Journal of Physics Research

Complex Isospectral Deformations of Conventional Shape-Invariant Superpotentials

Taha Koohrokhi, Abdolmajid Izadpanah

Published Sep 1, 2025• 0 citations
Publisher

Abstract

All additive shape-invariant superpotentials in nonrelativistic quantum mechanics can be classified into two categories: those that do not explicitly depend on ℏ and those that do. The latter category, known as the conventional superpotentials, forms a complete family. It has been demonstrated that the Schrödinger equation admits exact analytical solutions for this family, highlighting their significance in quantum mechanics due to this and other intriguing properties. This paper presents a mechanism for generalizing these superpotentials to the complex domain. The resulting complex non-Hermitian Hamiltonians possess real energy eigenvalues and are isospectral with their real counterparts..

DOI
10.47176/ijpr.25.2.22029

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All additive shape-invariant superpotentials in nonrelativistic quantum mechanics can be classified into two categories: those that do not explicitly depend on ℏ and those that do. The latter category, known as the conventional superpotentials, forms a complete family. It has been demonstrated that the Schrödinger equation admits exact analytical solutions for this family, highlighting their significance in quantum mechanics due to this and other intriguing properties. This paper presents a mechanism for generalizing these superpotentials to the complex domain. The resulting complex non-Hermitian Hamiltonians possess real energy eigenvalues and are isospectral with their real counterparts..
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.
We extend the standard intertwining relations used in supersymmetrical (SUSY) quantum mechanics which involve real superpotentials to complex superpotentials. This allows us to deal with a large class of non-Hermitian Hamiltonians and to study in general the isospectrality between complex potentials. In very specific cases we can construct in a natural way "quasicomplex" potentials which we define as complex potentials having a global property so as to lead to a Hamiltonian with real spectrum. We also obtained a class of complex transparent potentials whose Hamiltonian can be intertwined to a free Hamiltonian. We provide a variety of examples both for the radial problem (half axis) and for the standard one-dimensional problem (the whole axis), including remarks concerning scattering problems.
With a view to increasing the scope of applications of supersymmetric quantum mechanics (SUSY QM), we formulate the same in a complex phase space. Within this framework, the concept of shape invariance is reinvestigated and an insight into the eigenvalue spectra of non-Hermitian Hamiltonians is sought. The results are applied to a variety of potentials. We claim that the shape invariance for these potentials in the complex phase space can be retrieved in terms of the prescriptions already proposed in the conventional SUSY QM, in that the transformation of potential parameters takes the form of a reflection in the parameter space. Interestingly, some of these features turn out to be the generalization of the concept of quasi-parity used recently in the context of SUSY QM of PT-symmetric potentials. Further, a correspondence of the present approach with other complex formulations of SUSY QM and also in two real dimensions is demonstrated.
Anti-PT\mathcal{PT}PT Transformations And Complex Non-Hermitian PT\mathcal{PT}PT-Symmetric Superpartners
We propose a new algebraic formalism for constructing complex non-Hermitian PT\mathcal{PT}PT-symmetric superpartners by extending a conventional shape-invariant superpotential into the complex domain. The resulting potential is an unbroken super- and parity-time (PT\mathcal{PT}PT)-symmetric shape-invariant potential with real energy eigenvalues, maintaining this property for all parameter values. In order to restore the probabilistic interpretation within a true quantum theory, a new inner product called the CPT\mathcal{CPT}CPT-inner product is defined in PT\mathcal{PT}PT-symmetric quantum mechanics, replacing the Dirac Hermitian inner product. In this work, we propose a new version of the inner product called the anti-PT\mathcal{PT}PT (APT\mathcal{APT}APT)-inner product, ⟨A∣B⟩≡∣A⟩APT.∣B⟩\langle A|B\rangle\equiv |A\rangle^{\mathcal{APT}}.|B\rangle⟨A∣B⟩≡∣A⟩APT.∣B⟩, which replaces the previous versions without any additional considerations. This PT\mathcal{PT}PT-supersymmetric quantum mechanics framework also allows for the unification of various areas of physics, including classical optics and quantum mechanics. To validate the theory, we present exact solutions for optical waveguides and the quantum tunneling probability, demonstrating excellent agreement with experimental data for the probability of crossing the potential barrier in the 3H(d,n)4He\rm ^{3}H(d,n)^{4}He3H(d,n)4He reaction.
For gaining further insight into the nature of the eigenspectra of a complex octic potential [say], we investigate the quasi exact solutions of the Schr?dinger equation in an extended complex phase space characterized by . The analyticity property of the eigenfunction alone is found sufficient to throw light on the nature of eigenvalues and eigenfunction of a system. Explicit expressions of eigenvalues and eigenfunctions for the ground state as well as for the first excited state of a complex octic potential and its variant are worked out. It is found that imaginary part of the eigenvalue turns out to be zero for real coupling parameters, whereas it becomes non-zero for complex coupling parameters. However, the PT-symmetric version of a non-hermitian Hamiltonian possesses the real eigenvalue even if coupling parameters in the potential are complex.
The complex-valued quantum mechanics considers quantum motion on the complex plane instead of on the real axis, and studies the variations of a particle complex position, momentum and energy along a complex trajectory. On the basis of quantum Hamilton-Jacobi formalism in the complex space, we point out that having complex-valued motion is a universal property of quantum systems, because every quantum system is actually accompanied with an intrinsic complex Hamiltonian originating from the equation. It is revealed that the conventional real-valued quantum mechanics is a special case of the complex-valued quantum mechanics in that the eigen-structures of real and complex quantum systems, such as their eigenvalues, eigenfunctions and eigen-trajectories, are invariant under linear complex mapping. In other words, there is indeed no distinction between Hermitian systems, PT-symmetric systems, and non PT-symmetric systems when viewed from a complex domain. Their eigen-structures can be made coincident through linear transformation of complex coordinates.