Browsing by Author "Avinash Khare"

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A class of exactly solvable rationally extended Calogero–Wolfes type 3-body problems
(Academic Press Inc., 2017) Nisha Kumari; Rajesh Kumar Yadav; Avinash Khare; Bhabani Prasad Mandal
In this work, we start from the well known Calogero–Wolfes type 3-body problems on a line and construct the corresponding exactly solvable rationally extended 3-body potentials. In particular, we obtain the corresponding energy eigenvalues and eigenfunctions which are in terms of the product of Xm Laguerre and Xp Jacobi exceptional orthogonal polynomials where both m,p=1,2,3,. © 2017 Elsevier Inc.
A class of exactly solvable rationally extended non-central potentials in two and three dimensions
(American Institute of Physics Inc., 2018) Nisha Kumari; Rajesh Kumar Yadav; Avinash Khare; Bhabani Prasad Mandal
We start from a seven parameter (six continuous and one discrete) family of non-central exactly solvable potentials in three dimensions and construct a wide class of ten parameters (six continuous and four discrete) family of rationally extended exactly solvable non-central real as well as PT symmetric complex potentials. The energy eigenvalues and the eigenfunctions of these extended non-central potentials are obtained explicitly and it is shown that the wave eigenfunctions of these potentials are either associated with the exceptional orthogonal polynomials or some type of new polynomials which can be further re-expressed in terms of the corresponding classical orthogonal polynomials. Similarly, we also construct a wide class of rationally extended exactly solvable non-central real as well as complex PT-invariant potentials in two dimensions. © 2018 Author(s).
A class of exactly solvable real and complex PT symmetric reflectionless potentials
(American Institute of Physics Inc., 2024) Suman Banerjee; Rajesh Kumar Yadav; Avinash Khare; Bhabani Prasad Mandal
We consider the question of the number of exactly solvable complex but PT-invariant reflectionless potentials with N bound states. By carefully considering the Xm rationally extended reflectionless potentials, we argue that the total number of exactly solvable complex PT-invariant reflectionless potentials are 2[(2N − 1)m + N]. © 2024 Author(s).
A short note on “Group theoretic approach to rationally extended shape invariant potentials” [Ann. Phys. 359 (2015) 46–54]
(Academic Press Inc., 2017) Arturo Ramos; Bijan Bagchi; Avinash Khare; Nisha Kumari; Bhabani Prasad Mandal; Rajesh Kumar Yadav
It is proved the equivalence of the compatibility condition of Ramos (2011, 2012) with a condition found in Yadav et al. (2015). The link of Shape Invariance with the existence of a Potential Algebra is reinforced for the rationally extended Shape Invariant potentials. Some examples on X1 and Xℓ Jacobi and Laguerre cases are given. © 2017 Elsevier Inc.
Group theoretic approach to rationally extended shape invariant potentials
(Academic Press Inc., 2015) Rajesh Kumar Yadav; Nisha Kumari; Avinash Khare; Bhabani Prasad Mandal
The exact bound state spectrum of rationally extended shape invariant real as well as PT symmetric complex potentials is obtained by using potential group approach. The generators of the potential groups are modified by introducing a new operator U(x,J3±12)to express the Hamiltonian corresponding to these extended potentials in terms of Casimir operators. Connection between the potential algebra and the shape invariance is elucidated. © 2015 Elsevier Inc.
New quasi-exactly solvable hermitian as well as non-hermitian PT-invariant potentials
(2009) Avinash Khare; Bhabani Prasad Mandal
We start with quasi-exactly solvable (QES) Hermitian (and hence real) aswell as complex PT -invariant, double sinh-Gordon potential and show that even afteradding perturbation terms, the resulting potentials, in both cases, are still QES potentials. Further, by using anti-isospectral transformations, we obtain Hermitian as well as PT -invariant complex QES periodic potentials. We study in detail the various properties of the corresponding Bender-Dunne polynomials. ©Indian Academy of Sciences.
Novel symmetries in N=2 supersymmetric quantum mechanical models.
(2013) R.P. Malik; Avinash Khare
We demonstrate the existence of a novel set of discrete symmetries in the context of the N=2 supersymmetric (SUSY) quantum mechanical model with a potential function f(x) that is a generalization of the potential of the 1D SUSY harmonic oscillator. We perform the same exercise for the motion of a charged particle in the X-Y plane under the influence of a magnetic field in the Z-direction. We derive the underlying algebra of the existing continuous symmetry transformations (and corresponding conserved charges) and establish its relevance to the algebraic structures of the de Rham cohomological operators of differential geometry. We show that the discrete symmetry transformations of our present general theories correspond to the Hodge duality operation. Ultimately, we conjecture that any arbitrary N=2 SUSY quantum mechanical system can be shown to be a tractable model for the Hodge theory. © 2013 Elsevier Inc.
One parameter family of rationally extended isospectral potentials
(Academic Press Inc., 2022) Rajesh Kumar Yadav; Suman Banerjee; Nisha Kumari; Avinash Khare; Bhabani Prasad Mandal
We start from a given one dimensional rationally extended shape invariant potential associated with Xm exceptional orthogonal polynomials and using the idea of supersymmetry in quantum mechanics, we obtain one continuous parameter (λ) family of rationally extended strictly isospectral potentials. We illustrate this construction by considering three well known rationally extended potentials, two with pure discrete spectrum (the extended radial oscillator and the extended Scarf-I) and one with both the discrete and the continuous spectrum (the extended generalized Pöschl–Teller) and explicitly construct the corresponding one continuous parameter family of rationally extended strictly isospectral potentials. Further, in the special case of λ=0 and −1, we obtain two new exactly solvable rationally extended potentials, namely the rationally extended Pursey and the rationally extended Abraham–Moses potentials respectively. We illustrate the whole procedure by discussing in detail the particular case of the X1 rationally extended one parameter family of potentials including the corresponding Pursey and the Abraham Moses potentials. © 2021 Elsevier Inc.
Parametric symmetries in exactly solvable real and PT symmetric complex potentials
(American Institute of Physics Inc., 2016) Rajesh Kumar Yadav; Avinash Khare; Bijan Bagchi; Nisha Kumari; Bhabani Prasad Mandal
In this paper, we discuss the parametric symmetries in different exactly solvable systems characterized by real or complex PT symmetric potentials. We focus our attention on the conventional potentials such as the generalized Pöschl Teller (GPT), Scarf-I, and PT symmetric Scarf-II which are invariant under certain parametric transformations. The resulting set of potentials is shown to yield a completely different behavior of the bound state solutions. Further, the supersymmetric partner potentials acquire different forms under such parametric transformations leading to new sets of exactly solvable real and PT symmetric complex potentials. These potentials are also observed to be shape invariant (SI) in nature.We subsequently take up a study of the newly discovered rationally extended SI potentials, corresponding to the above mentioned conventional potentials, whose bound state solutions are associated with the exceptional orthogonal polynomials (EOPs).We discuss the transformations of the corresponding Casimir operator employing the properties of the so(2,1) algebra.
Rationally extended many-body truncated Calogero–Sutherland model
(Academic Press Inc., 2019) Rajesh Kumar Yadav; Avinash Khare; Nisha Kumari; Bhabani Prasad Mandal
We construct a rational extension of the truncated Calogero–Sutherland model by Pittman et al. The exact solution of this rationally extended model is obtained analytically and it is shown that while the energy eigenvalues remain unchanged, however the eigenfunctions are completely different and written in terms of exceptional X1 Laguerre orthogonal polynomials. The rational model is further extended to a more general Xm case by introducing m dependent interaction term. As expected, in the special case of m = 0, the extended model reduces to the conventional model of Pittman et al. In the two appropriate limits, we thereby obtain rational extensions of the celebrated Calogero–Sutherland as well as Jain–Khare models. The multi-index extension of the model is also discussed. © 2018 Elsevier Inc.
Rationally extended shape invariant potentials in arbitrary d dimensions associated with exceptional Xm polynomials
(Czech Technical University in Prague, 2017) Rajesh Kumar Yadav; Nisha Kumari; Avinash Khare; Bhabani Prasad Mandal
Rationally extended shape invariant potentials in arbitrary D-dimensions are obtained by using point canonical transformation (PCT) method. The bound-state solutions of these exactly solvable potentials can be written in terms of Xm Laguerre or Xm Jacobi exceptional orthogonal polynomials. These potentials are isospectral to their usual counterparts and possess translationally shape invariance property. © Czech Technical University in Prague, 2017.
Scattering amplitudes for the rationally extended PT symmetric complex potentials
(Academic Press Inc., 2016) Nisha Kumari; Rajesh Kumar Yadav; Avinash Khare; Bijan Bagchi; Bhabani Prasad Mandal
In this paper, we consider the rational extensions of two different classes of PT symmetric complex potentials namely the asymptotically vanishing Scarf II and asymptotically non-vanishing Rosen–Morse II [ RM-II] and obtain the accompanying bound state eigenfunctions in terms of the exceptional Xm Jacobi polynomials and a certain class of orthogonal polynomials. By considering the asymptotic behavior of the exceptional polynomials, we also derive the reflection and transmission amplitudes for them and discuss the various novel properties of the corresponding amplitudes. © 2016 Elsevier Inc.
Solutions of one-dimensional Dirac equation associated with exceptional orthogonal polynomials and the parametric symmetry
(World Scientific, 2023) Suman Banerjee; Rajesh Kumar Yadav; Avinash Khare; Nisha Kumari; Bhabani Prasad Mandal
We consider one-dimensional Dirac equation with rationally extended scalar potentials corresponding to the radial oscillator, the trigonometric Scarf and the hyperbolic Pöschl-Teller potentials and obtain their solution in terms of exceptional orthogonal polynomials. Further, in the case of the trigonometric Scarf and the hyperbolic Pöschl-Teller cases, a new family of Dirac scalar potentials is generated using the idea of parametric symmetry and their solutions are obtained in terms of conventional as well as exceptional orthogonal polynomials. © 2023 World Scientific Publishing Company.
Supersymmetry and shape invariance of exceptional orthogonal polynomials
(Academic Press Inc., 2022) Satish Yadav; Avinash Khare; Bhabani Prasad Mandal
We discuss the exceptional Laguerre and the exceptional Jacobi orthogonal polynomials in the framework of the supersymmetric quantum mechanics (SUSYQM). We express the differential equations for the Jacobi and the Laguerre exceptional orthogonal polynomials (EOP) as the eigenvalue equations and make an analogy with the time independent Schrödinger equation to define “Hamiltonians” enables us to study the EOPs in the framework of the SUSYQM and to realize the underlying shape invariance associated with such systems. We show that the underlying shape invariance symmetry is responsible for the solubility of the differential equations associated with these polynomials. © 2022 Elsevier Inc.
The scattering amplitude for a newly found exactly solvable potential
(2013) Rajesh Kumar Yadav; Avinash Khare; Bhabani Prasad Mandal
The scattering amplitude for a recently discovered exactly solvable shape invariant potential, which is isospectral to the generalized Pöschl-Teller potential, is calculated explicitly by considering the asymptotic behavior of the X1 Jacobi exceptional polynomials associated with this system. © 2013 Elsevier Inc.
The scattering amplitude for one parameter family of shape invariant potentials related to Xm jacobi polynomials
(2013) Rajesh Kumar Yadav; Avinash Khare; Bhabani Prasad Mandal
We consider the recently discovered, one parameter family of exactly solvable shape invariant potentials which are isospectral to the generalized Pöschl-Teller potential. By explicitly considering the asymptotic behavior of the Xm Jacobi polynomials associated with this system (m=1, 2, 3, ...), the scattering amplitude for the one parameter family of potentials is calculated explicitly. © 2013 Elsevier B.V.
The scattering amplitude for rationally extended shape invariant Eckart potentials
(Elsevier B.V., 2015) Rajesh Kumar Yadav; Avinash Khare; Bhabani Prasad Mandal
We consider the rationally extended exactly solvable Eckart potentials which exhibit extended shape invariance property. These potentials are isospectral to the conventional Eckart potential. The scattering amplitude for these rationally extended potentials is calculated analytically for the generalized mth (m=1,2,3,.) case by considering the asymptotic behavior of the scattering state wave functions which are written in terms of some new polynomials related to the Jacobi polynomials. As expected, in the m=0 limit, this scattering amplitude goes over to the scattering amplitude for the conventional Eckart potential. © 2014 Published by Elsevier B.V.