Shape Invariance Symmetry Research Articles

Supersymmetry and shape invariance of exceptional orthogonal polynomials

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.

New research based on the new high-order generalized uncertainty principle for Klein–Gordon equation

A new high-order generalized uncertainty principle is proposed in this paper, which can modify the coordinate operator and the momentum operator simultaneously. Afterwards, the Klein–Gordon equation with linear scalar and vector potential is investigated in the context of new principle and whose corresponding exact analytical solutions are further derived. In addition, the above results are also verified in a pure algebraic fashion by using the property of shape invariance symmetry. Moreover, the influence of the minimum length parameter β on the energy spectrum of the Klein–Gordon equation is also discussed in detail in this article.

Deformed shape invariance symmetry and potentials in curved space with two known eigenstates

We consider two families of extensions of the oscillator in a d-dimensional constant-curvature space and analyze them in a deformed supersymmetric framework, wherein the starting oscillator is known to exhibit a deformed shape invariance property. We show that the first two members of each extension family are also endowed with such a property, provided some constraint conditions relating the potential parameters are satisfied, in other words they are conditionally deformed shape invariant. Since, in the second step of the construction of a partner potential hierarchy, the constraint conditions change, we impose compatibility conditions between the two sets to build potentials with known ground and first excited states. To extend such results to any members of the two families, we devise a general method wherein the first two superpotentials, the first two partner potentials, and the first two eigenstates of the starting potential are built from some generating function W+(r) [and its accompanying function W−(r)].

Application of conditional shape invariance symmetry to obtain the eigen-spectrum of the mixed potential [formula omitted

We show that the conditional shape invariance symmetry can be used as a very powerful tool to calculate the eigenvalues of the mixed potential V(r)=ar+br2+cr+l(l+1)r2 for a restricted set of potential parameters. The energy for any state can be obtained algebraically, albeit for a severely restricted set of potential parameters. We also indicate that each member of the hierarchy of Hamiltonians is basically conditionally translational shape invariant. Comparison of analytically obtained results with numerical results is also presented. Our present methodology can be taken as an alternative treatment for the calculation of any higher order excited states of conditionally exactly solvable (CES) potentials.

The role of shape invariance potentials in the relativistic quantum mechanics

The point canonical transformation in non-relativistic quantum mechanics is applied as an algebraic method to obtain the solutions of the Dirac equation with spherical symmetry electromagnetic potentials. We want to show that some of the non-relativistic solvable potentials with shape-invariant symmetry can be related to the radial Dirac equation. Using this method, the idea of supersymmetry and shape invariance can be expanded to the relativistic quantum mechanics. The spinor wave functions for some of the obtained four-component electromagnetic potential are given in terms of special functions such as Jacobi, generalized Laguerre and Hermite polynomials. The relativistic bound-states spectrum for each case is also calculated in terms of the bound-states spectrum of the solvable potentials.

Infinite families of (non)-Hermitian Hamiltonians associated with exceptional Xm Jacobi polynomials

Using an appropriate change of variable, the Schrödinger equation is transformed into a second-order differential equation satisfied by recently discovered Jacobi-type Xm exceptional orthogonal polynomials. This facilitates the derivation of infinite families of exactly solvable Hermitian as well as non-Hermitian trigonometric Scarf potentials and a finite number of Hermitian and an infinite number of non-Hermitian -symmetric hyperbolic Scarf potentials. The bound state solutions of all these potentials are associated with the aforesaid exceptional orthogonal polynomials. These infinite families of potentials are shown to be extensions of the conventional trigonometric and hyperbolic Scarf potentials by the addition of some rational terms characterized by the presence of classical Jacobi polynomials. All the members of a particular family of these ‘rationally extended polynomial-dependent’ potentials have the same energy spectrum and possess translational shape-invariant symmetry. The obtained non-Hermitian trigonometric Scarf potentials are shown to be quasi-Hermitian in nature ensuring the reality of the associated energy spectra.

Calculation of state energy of (2n+1)-fold wells using the spectral properties of supersymmetry shape-invariant potential

Shape invariance is an important factor of many exactly solvable quantum mechanics. Several examples of shape-invariant ‘discrete quantum mechanical systems’ are introduced and discussed in some detail. We present the spectral properties of supersymmetric shape-invariant potentials (SIP). Here we are interested in some time-independent integrable systems which are exactly solvable owing to the existence of supersymmetric shape-invariant symmetry. In 1981 Witten proposed (0+1)-dimensional limit of supersymmetry (SUSY) quantum field theory, where the supercharges of SUSY quantum mechanics generate transformation between two orthogonal eigenstates of a given Hamiltonian wit degenerate eigenvaluesfor the non-SIP as very few lower eigenvalues can be known analytically, which are small to calculate spectral fluctuation.

Supersymmetry, shape invariance and the Legendre equations

In three space dimensions, when a physical system possesses spherical symmetry, the dynamical equations automatically lead to the Legendre and the associated Legendre equations, with the respective orthogonal polynomials as their standard solutions. This is a very general and important result and appears in many problems in physics (for example, the multipole expansion, etc.). We study these equations from an operator point of view, much like the harmonic oscillator, and show that there is an underlying shape invariance symmetry in these systems responsible for their solubility. We bring out various interesting features resulting from this analysis from the shape invariance point of view.

Dynamical and structural symmetries for the highest Landau levels on the AdS 2

Abstract The aim of the paper is to use the recurrence relations with respect to both indices of the associated Legendre functions for the extraction of the Dirac quantization condition and dynamical symmetry group U(1, 1) corresponding to the highest Landau levels on the hyperbolic plane with uniform magnetic field B. Irreducible representations of the su(2) algebra are obtained by the ladder differential operators which change B by 1/2 unit and mode number by one unit. Two different classes of the irreducible representations of SU(1, 1) with the even and odd boson numbers 2B − 1/2 are extracted for the Bargmann indices 1/4 and 3/4, respectively. Finally, we show that shape invariance symmetry is realized by the ladder operators which shift only the magnetic field B by 1/2 unit.

Continuous Spectrum of Trigonometric Rosen-Morse and Eckart Potentials from Free Particle Spectrum

The shape invariant symmetry of the Trigonometric Rosen-Morse and Eckart potentials has been studied through realization of so(3) and so(2,1) Lie algebras respectively. In this work, by using the free particle eigenfunction, we obtain continuous spectrum of these potentials by means of their shape invariance symmetry in an algebraic method.

Nonlinear phenomena, optical and quantum solitons

Research in the theory of solitons and its applications has undergone impressive development in recent years and has affected a broad range of fields, including theory of special functions, quantum integrable systems, numerical analysis, cellular automata, representations of quantum groups and

Shape Invariant and Rodrigues Solution of the Dirac-shifted Oscillator and Dirac-Morse Potentials

We show that the Dirac equation for a charged spinor in spherically symmetric electromagnetic potentials as Dirac-shifted oscillator and Dirac-Morse potentials have the conditions of shape invariant symmetry in non-relativistic quantum mechanics. The relativistic spectra of the bound states and spinor wavefunctions can be obtained by the Rodrigues polynomials of one associated differential equation.

A generalized quantum nonlinear oscillator

We examine various generalizations, e.g. exactly solvable, quasi-exactly solvable and non-Hermitian variants, of a quantum nonlinear oscillator. For all these cases, the same mass function has been used and it has also been shown that the new exactly solvable potentials possess shape invariance symmetry. The solutions are obtained in terms of classical orthogonal polynomials.

Energy eigenvalues of a quantum anharmonic oscillator from supersymmetry: the concept of conditional shape-invariance symmetry

The eigenvalues for oscillators with quartic and sextic anharmonicities have been calculated for a restricted set of parameters by supersymmetric quantum mechanics. We show that energies of at most a few levels can be obtained algebraically for a severely restricted set of parameters resulting from the satisfaction of a few constraint conditions. We present how conditional shape-invariance symmetry of supersymmetric partner potentials leads to the conditional exact solution. We also indicate its generalization in N dimensions.

Two-Dimensional Quantum Hamiltonians with Shape Invariance Symmetry

It is shown that the Casimir operator associated with the U(1) Lie derivative defined on the S2=SU(2)/U(1) base manifold, can be interpreted as Hamiltonians of a pair of scalar particle and scalar anti-particle with opposite charges over the S2 manifold in the presence of a magnetic monopole located at its origin and an external electric field. Using the SU(2) representation, the spectra of these Hamiltonians have been obtained. It is also proved that these Hamiltonians are isospectral and having the shape invariance symmetry, i.e. they are supersymmetric partner of each other. Also the Dirac’s quantization of magnetic charge comes very naturally from the finiteness of the SU(2) representation.

Spectral properties of supersymmetric shape invariant potentials

We present the spectral properties of supersymmetric shape invariant potentials (SIPs). Although the folded spectrum is completely random, unfolded spectrum shows that energy levels are highly correlated and absolutely rigid. All the SIPs exhibit harmonic oscillator-type spectral statistics in the unfolded spectrum. We conjecture that this is the reflection of shape invariant symmetry.

Ladder operators and recursion relations for the associated Bessel polynomials

Introducing the associated Bessel polynomials in terms of two non-negative integers, and under an integrability condition we simultaneously factorize their corresponding differential equation into a product of the ladder operators by four different ways as shape invariance symmetry equations. This procedure gives four different pairs of recursion relations on the associated Bessel polynomials. In spite of description of Bessel and Laguerre polynomials in terms of each other, we show that the associated Bessel differential equation is factorized in four different ways whereas for Laguerre one we have three different ways.

REPRESENTATION OF THE HEISENBERG ALGEBRA h4 BY THE LOWEST LANDAU LEVELS AND THEIR COHERENT STATES

Using simultaneous shape invariance with respect to two different parameters, we introduce a pair of appropriate operators which realize shape invariance symmetry for the monomials on a half-axis. It leads to the derivation of rotational symmetry and dynamical symmetry group H4 with infinite-fold degeneracy for the lowest Landau levels. This allows us to represent the Heisenberg–Lie algebra h4 not only by the lowest Landau levels, but also by their corresponding standard coherent states.

Two and three dimensional Hamiltonians with generalized and ordinary shape invariance symmetry

Two and three dimensional Hamiltonians with generalized and ordinary shape invariance symmetry have been obtained by Fourier transforming over some coordinates of the SU(3) Casimir operator defined on SU(3)∕SU(2) symmetric space. It is shown that the generalized shape invariance of the two dimensional Hamiltonian is equivalent to SU(3) symmetry while in the three dimensional one, the ordinary shape invariance is equivalent to contracted SU(3) and there is one to one correspondence between the representations of the generalized shape invariance symmetry of the two (three) dimensional Hamiltonian and SU(3) [contracted SU(3)] Verma bases.

Shape invariance symmetries for quantum states of the superpotentials Atanh ωy+ B/ A and − Acot ωθ+ Bcsc ωθ

Using shape invariance symmetries with respect to two different parameters n and m, we derive the general forms of the partner Hamiltonians for the superpotentials Atanh ωy+ B/ A and − Acot ωθ+ Bcsc ωθ, respectively. For the first model in a special case and the second model in the general case, the parameters m and n play the role of the quantization numbers for the spectrum and quantum states as finite and infinite ones, respectively. We also get a type of the shape invariance symmetry which is realized by the operators shifting only m and only n, respectively, for two models. Furthermore, for the models, other shape invariance symmetries based on the operators shifting the indices n and m of quantum states simultaneously and inversely as well as simultaneously and agreeably are derived.