A brief overview is given of recent developments and fresh ideas at the intersection of [Formula: see text]- and/or [Formula: see text]-symmetric quantum mechanics with supersymmetric quantum mechanics (SUSY QM). Within the framework of the resulting supersymmetric version of [Formula: see text]-symmetric quantum mechanics we study the consequences of the assumption that the "charge" operator [Formula: see text] is represented in a differential-operator form of the second or higher order. Besides the freedom allowed by the Hermiticity constraint for the operator [Formula: see text], encouraging results are obtained in the second-order case. In particular, the integrability of intertwining relations proves to match the closure of our nonlinear (viz., polynomial) SUSY algebra. In a particular illustration, our form of [Formula: see text]-symmetric SUSY QM leads to a new class of non-Hermitian polynomial oscillators with real spectrum which turn out to be [Formula: see text]-asymmetric.
CPT-CONSERVING HAMILTONIANS AND THEIR NONLINEAR SUPERSYMMETRIZATION USING DIFFERENTIAL CHARGE-OPERATORS C
A brief overview is given of recent developments and fresh ideas at the intersection of [Formula: see text]- and/or [Formula: see text]-symmetric quantum mechanics with supersymmetric quantum mechanics (SUSY QM). Within the framework of the resulting supersymmetric version of [Formula: see text]-symmetric quantum mechanics we study the consequences of the assumption that the "charge" operator [Formula: see text] is represented in a differential-operator form of the second or higher order. Besides the freedom allowed by the Hermiticity constraint for the operator [Formula: see text], encouraging results are obtained in the second-order case. In particular, the integrability of intertwining relations proves to match the closure of our nonlinear (viz., polynomial) SUSY algebra. In a particular illustration, our form of [Formula: see text]-symmetric SUSY QM leads to a new class of non-Hermitian polynomial oscillators with real spectrum which turn out to be [Formula: see text]-asymmetric.
We review the higher‐order supersymmetric quantum mechanics (H‐SUSY QM), which involves differential intertwining operators of order greater than one. The iterations of first‐order SUSY transformations are used to derive in a simple way the higher‐order case. The second order technique is addressed directly, and through this approach unexpected possibilities for designing spectra are uncovered. The formalism is applied to the harmonic oscillator: the corresponding H‐SUSY partner Hamiltonians are ruled by polynomial Heisenberg algebras which allow a straight construction of the coherent states.
The study of the non-Hermitian Hamiltonian systems within the framework of Supersymmetric Quantum mechanics (SUSY QM) has been carried out in literature in the last decade. Most of these potentials are either PT Symmetric or quasi-Hermitian in nature. But no formulation is advanced in literature to the study of general complex potentials within its framework. In this paper, using the complex phase space approach, we carry out the complexification of Supersymmetric Quantum Mechanics to enable the study of any general non-Hermitian Hamiltonian system within its framework. The commutation relations among its operators are enumerated and the underlying symmetry is discussed. The eigenvalues and eigenfunctions of the real and imaginary parts of superpartner Hamiltonians are also investigated.
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.
PT Symmetric Hamiltonian Model and Exactly Solvable Potentials
Searching for non-Hermitian (parity-time)PT-symmetric Hamiltonians \cite{bender} with real spectra has been acquiring much interest for fourteen years. In this article, we have introduced a PT symmetric non-Hermitian Hamiltonian model which is given as H^=ω(b^†b^+21)+α(b^2−(b^†)2) where ω and α are real constants, b^ and b†^ are first order differential operators. Moreover, Pseudo-Hermiticity that is a generalization of PT symmetry has been attracting a growing interest \cite{mos}. Because the Hamiltonian H is pseudo-Hermitian, we have obtained the Hermitian equivalent of 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 is called pseudo-Hermitian, if there exists a Hermitian and invertible operator η satisfying H†=ηHη−1. For the Hermitian Hamiltonian h, one can write h=ρHρ−1 where ρ=η is unitary. Using this ρ we have obtained a physical Hamiltonian h for each case. Then, the Schrödinger equation is solved exactly using Shape Invariance method of Supersymmetric Quantum Mechanics \cite{susy1}. Mapping function ρ is obtained for each potential case.
LINKED PT-SYMMETRY TO SUPERSYMMETRY IN A CLASS OF NON-HERMITIAN HAMILTONIANS
We introduce and study a class of non-Hermitian Hamiltonians which have velocity dependent potentials. Since stability cannot 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-[Formula: see text]-symmetric Hamiltonians can be transformed into their corresponding superpartner Hamiltonians via a specific canonical transformation while the [Formula: see text]-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-[Formula: see text]-symmetric Hamiltonians out of the introduced class of Hamiltonians have the same spectrum as the corresponding superpartner Hamiltonians and thus supersymmetry (Susy) is broken for such Hamiltonians. This kind of intertwining of [Formula: see text]-symmetry and Susy is new as all the so far discussed cases concentrate on Hamiltonians of broken [Formula: see text]-symmetry that have broken Susy too while we showed that Susy can be also broken for non-[Formula: see text]-symmetric and non-Hermitian Hamiltonians.
By modifying and generalizing known supersymmetric models, we are able to find four different sets of one-dimensional Hamiltonians for the inverted harmonic oscillator. The first set of Hamiltonians is derived by extending the supersymmetric quantum mechanics with reflections to non-Hermitian supercharges. The second set is obtained by generalizing the supersymmetric quantum mechanics valid for non-Hermitian supercharges with the Dunkl derivative instead of [Formula: see text]. Also, by changing the derivative [Formula: see text] by the Dunkl derivative in the creation and annihilation-type operators of the standard inverted harmonic oscillator [Formula: see text], we generate the third set of Hamiltonians. The fourth set of Hamiltonians emerges by allowing a parameter of the supersymmetric two-body Calogero-type model to take imaginary values. The eigensolutions of definite parity for each set of Hamiltonians are given.