Shape Invariance Property Research Articles

‘Diophantine’ and factorization properties of finite orthogonal polynomials in the Askey scheme

A new interpretation and applications of the ‘Diophantine’ and factorization properties of finite orthogonal polynomials in the Askey scheme are explored. The corresponding twelve polynomials are the (q-)Racah, (dual, q-)Hahn, Krawtchouk and five types of q-Krawtchouk. These (q-)hypergeometric polynomials, defined only for the degrees of 0 , 1 , … , N , constitute the main part of the eigenvectors of N + 1-dimensional tri-diagonal real symmetric matrices, which correspond to the difference equations governing the polynomials. The monic versions of these polynomials all exhibit the ‘Diophantine’ and factorization properties at higher degrees than N. This simply means that these higher degree polynomials are zero-norm ‘eigenvectors’ of the N + 1-dimensional tri-diagonal real symmetric matrices. A new type of multi-indexed orthogonal polynomials belonging to these twelve polynomials could be introduced by using the higher degree polynomials as the seed solutions of the multiple Darboux transformations for the corresponding matrix eigenvalue problems. The shape-invariance properties of the simplest type of the multi-indexed polynomials are demonstrated. The explicit transformation formulas are presented.

Design and generation of structured array beams with shape-invariant properties

We theoretically propose and experimentally demonstrate a flexible approach to generate an arbitrary shape-invariant structured array beam during propagation with the prescribed intensity distribution. An algorithm for generating flexible desired phase is presented according the reverse engineering and Huygens–Fresnel theory. In this paper, we outline the theory for creating the shape-invariant array beams. The numerical simulation results show that the shape of generated array beams is invariant and its size increases gradually from the near field to the far field. The generation of shape-invariant structured array beams can be realized by the phase-only and amplitude-only elements encoded on the spatial light modulators. The generated arbitrary shape-invariant structured array beams are verified and are in excellent agreement with the theory.

Quasi-exactly solvable extensions of the Kepler–Coulomb potential on the sphere

We consider a family of extensions of the Kepler–Coulomb potential on a d-dimensional sphere and analyze it in a deformed supersymmetric framework, wherein the starting potential is known to exhibit a deformed shape invariance property. We show that the members of the extended 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 quasi-exactly solvable potentials with known ground and first-excited states. Some explicit results are obtained for the first three members of the family. We then use a generating function 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)]. From the results obtained for the latter for the first three family members, we propose some formulas for W±(r) valid for the mth family member, depending on m+1 constants a0, a1, …, am. Such constants satisfy a system of m+1 linear equations. Solving the latter allows us to extend the results up to the seventh family member and then to formulate a conjecture giving the general structure of the ai constants in terms of the parameters of the problem.

Shape Invariant Potentials in Supersymmetric Quantum Cosmology

In this brief review, we comment on the concept of shape invariant potentials, which is an essential feature in many settings of N=2 supersymmetric quantum mechanics. To motivate its application within supersymmetric quantum cosmology, we present a case study to illustrate the value of this promising tool. Concretely, we take a spatially flat FRW model in the presence of a single scalar field, minimally coupled to gravity. Then, we extract the associated Schrödinger–Wheeler–DeWitt equation, allowing for a particular scope of factor ordering. Subsequently, we compute the corresponding supersymmetric partner Hamiltonians, H1 and H2. Moreover, we point out how the shape invariance property can be employed to bring a relation among several factor orderings choices for our Schrödinger–Wheeler–DeWitt equation. The ground state is retrieved, and the excited states easily written. Finally, the Hamiltonians, H1 and H2, are explicitly presented within a N=2 supersymmetric quantum mechanics framework.

Ladder operators and coherent states for the Rosen–Morse system and its rational extensions

Ladder operators for the hyperbolic Rosen–Morse (RMII) potential are realized using the shape invariance property appearing, in particular, using supersymmetric quantum mechanics. The extension of the ladder operators to a specific class of rational extensions of the RMII potential is presented and discussed. Coherent states are then constructed as almost eigenstates of the lowering operators. Some properties are analyzed and compared. The ladder operators and coherent states constructions presented are extended to the case of the trigonometric Rosen–Morse (RMI) potential using a point canonical transformation.

Exactly solvable new classes of potentials with finite discrete energies

In this work, we propose more realistic models with discrete and finite number of energy levels that could fit well to molecules with potentials that were modeled previously as harmonic oscillator. The considered potentials could be also used as good models in quantum physics, statistical and condensed matter physics, atomic physics, nuclear physics, particle physics, high energy physics, mathematical physics, as well as in chemistry of complex molecules. More precisely, we derive the solutions of two families of Schrödinger equations using supersymmetric quantum mechanics technique for superpotentials having shape invariance properties, and where their eigenvalues and eigenfunctions are exactly determined. The range of their finite number of bound states is given explicitly. Furthermore, this result will contribute in extending the already small list of exactly solvable Schrödinger equations, where we have summarized in a table all well-known potentials having exact solutions and their superpotentials, their partner potentials, and their energies, as well as, the newly discovered potentials proposed here.

Revisiting generalized Hulthén potentials

A relation between the deformed Hulthen potential and the Eckart one is used to write the bound-state wavefunctions of the former in terms of Jacobi polynomials and to calculate their normalization coefficients. The shape invariance property of the Eckart potential in standard first-order supersymmetric quantum mechanics allows to easily rederive the set of extended deformed Hulthen potentials, recently obtained by using the Darboux–Crum transformation, and to show that their spectra and normalized wavefunctions follow without any further calculation. The present approach considerably simplifies the previous derivation. Furthermore, by taking advantage of other known rational extensions of the Eckart potential obtained in first-order supersymmetric quantum mechanics, novel extensions of the deformed Hulthen potential are constructed, together with their bound-state spectra and wavefunctions. These new extensions belong to three different types, the first two being isospectral to some previously obtained extensions and the third one with an extra bound state below their spectrum.

Stability of kinklike structures in generalized models

We study the stability of topological structures in generalized models with a single real scalar field. We show that it is driven by a Sturm-Liouville equation and investigate the conditions that lead to the existence of explicit supersymmetric operators that factorize the stability equation and allow us to construct partner potentials. In this context, we discuss the property of shape invariance as a possible manner to calculate the discrete states and their respective eigenvalues.

Quasi-exactly solvable extended trigonometric Pöschl-Teller potentials with position-dependent mass

Infinite families of quasi-exactly solvable position-dependent mass Schrodinger equations with known ground and first excited states are constructed in a deformed supersymmetric background. The starting points consist in one- and two-parameter trigonometric Poschl-Teller potentials endowed with a deformed shape invariance property and, therefore, exactly solvable. Some extensions of them are considered with the same position-dependent mass and dealt with by a generating function method. The latter enables to construct the first two superpotentials of a deformed supersymmetric hierarchy, as well as the first two partner potentials and the first two eigenstates of the first potential from some generating function W+(x) (and its accompanying function W-(x) . The generalized trigonometric Poschl-Teller potentials so obtained are thought to have interesting applications in molecular and solid state physics.

Infinite families of position-dependent mass Schrödinger equations with known ground and first excited states

A construction method of infinite families of quasi-exactly solvable position-dependent mass Schrödinger equations with known ground and first excited states is proposed in a deformed supersymmetric background. Such families correspond to extensions of known potentials endowed with a deformed shape invariance property. Two different approaches are combined. The first one is a generating function method, which enables to construct the first two superpotentials of a deformed supersymmetric hierarchy, as well as the first two partner potentials and the first two eigenstates of the first potential, from some generating function W+(x) [and its accompanying function W−(x)]. The second approach is the conditionally deformed shape invariance method, wherein the deformed shape invariance property of the starting potentials is generalized to their extensions by adding some constraints on the parameters and by imposing compatibility conditions between sets of constraints. Detailed results are given for some extensions of the linear and radial harmonic oscillators, as well as the Kepler–Coulomb and Morse potentials.

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)].

RATIONALLY EXTENDED SHAPE INVARIANT POTENTIALS IN ARBITRARY D DIMENSIONS ASSOCIATED WITH EXCEPTIONAL Xm POLYNOMIALS

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 <em>X<sub>m</sub></em> Laguerre or <em>X<sub>m</sub></em> Jacobi exceptional orthogonal polynomials. These potentials are isospectral to their usual counterparts and possess translationally shape invariance property.

Sharpening enhancement technique for MR images to enhance the segmentation

Abstract This study presents a sharpening image enhancement technique based on the Laplacian pyramid (LP) and singular value decomposition (SVD) to improve the visibility and segmentation of subtle organs. The technique utilizes the shape-invariant properties of LP and the SVD techniques to enhance the perceptual sharpness of an image. The sharpening enhancement of magnetic resonance images not only sharpens the edges, but also reduces the noise effect. The results are compared with state-of-art enhancement techniques. The performance measures like peak-signal-to noise (PSNR), mean structural similarity index (MSSIM), and absolute mean brightness error (AMBE) evaluates the improved performance of the proposed technique.

New determinant expressions of multi-indexed orthogonal polynomials in discrete quantum mechanics

The multi-indexed orthogonal polynomials (the Meixner, little $q$-Jacobi (Laguerre), ($q$-)Racah, Wilson, Askey-Wilson types) satisfying second order difference equations were constructed in discrete quantum mechanics. They are polynomials in the sinusoidal coordinates $\eta(x)$ ($x$ is the coordinate of quantum system) and expressed in terms of the Casorati determinants whose matrix elements are functions of $x$ at various points. By using shape invariance properties, we derive various equivalent determinant expressions, especially those whose matrix elements are functions of the same point $x$. Except for the ($q$-)Racah case, they can be expressed in terms of $\eta$ only, without explicit $x$-dependence.

Generalized Coherent States for Position-Dependent Effective Mass Systems

A generalized scheme for the construction of coherent states in the context of position-dependent effective mass systems has been presented. This formalism is based on the ladder operators and associated algebra of the system which are obtained using the concepts of supersymmetric quantum mechanics and the property of shape invariance. In order to exemplify the general results and to analyze the properties of the coherent states, several examples have been considered.

Quantum oscillator and Kepler–Coulomb problems in curved spaces: Deformed shape invariance, point canonical transformations, and rational extensions

The quantum oscillator and Kepler-Coulomb problems in d-dimensional spaces with constant curvature are analyzed from several viewpoints. In a deformed supersymmetric framework, the corresponding nonlinear potentials are shown to exhibit a deformed shape invariance property. By using the point canonical transformation method, the two deformed Schrödinger equations are mapped onto conventional ones corresponding to some shape-invariant potentials, whose rational extensions are well known. The inverse point canonical transformations then provide some rational extensions of the oscillator and Kepler-Coulomb potentials in curved space. The oscillator on the sphere and the Kepler-Coulomb potential in a hyperbolic space are studied in detail and their extensions are proved to be consistent with already known ones in Euclidean space. The partnership between nonextended and extended potentials is interpreted in a deformed supersymmetric framework. Those extended potentials that are isospectral to some nonextended ones are shown to display deformed shape invariance, which in the Kepler-Coulomb case is enlarged by also translating the degree of the polynomial arising in the rational part denominator.

Analytical Solution of Dirac Equation for q-Deformed Hyperbolic Manning-Rosen Potential in D Dimensions using SUSY QM and its Thermodynamics'Application

The Dirac equation of q-deformed hyperbolic Manning Rosen potential in D dimension was solved by using Supersymmetric Quantum Mechanics (SUSY QM). The D dimensional relativistic energy spectra were obtained by using SUSY QM and shape invariant properties and D dimensional wave functions of q-deformed hyperbolic Manning Rosen potential were obtained by using the SUSY raising and lowering operators. In the nonrelativistic limit, the relativistic energy spectra for exact spin symmetry case reduced into nonrelativistic energy spectra and so for the wave functions. In the classical regime, the partition function, the vibrational specific heat, and the vibrational mean energy of some diatomic molecules were calculated from the non-relativistic energy spectra with the help of error function and imaginary error function.

Solution of The Schrödinger Equation for Trigonometric Scarf Plus Poschl-Teller Non-Central Potential Using Supersymmetry Quantum Mechanics

<span>In this paper, we show that the exact energy eigenvalues and eigen functions of the Schrödinger <span>equation for charged particles moving in certain class of noncentral potentials can be easily <span>calculated analytically in a simple and elegant manner by using Supersymmetric method <span>(SUSYQM). We discuss the trigonometric Scarf plus Poschl-Teller systems. Then, by operating <span>the lowering operator we get the ground state wave function, and the excited state wave functions <span>are obtained by operating raising operator repeatedly. The energy eigenvalue is expressed in the <span>closed form obtained using the shape invariant properties. The results are in exact agreement with <span>other methods.</span></span></span></span></span></span></span><br /></span>

Thermodynamics properties study of diatomic molecules with q-deformed modified Poschl-Teller plus Manning Rosen non-central potential in D dimensions using SUSYQM approach

D-dimensional Dirac equation of q-deformed modified Poschl-Teller plus Manning Rosen non-central potential was solved using supersymmetric quantum mechanics (SUSY QM). The relativistic energy spectra were analyzed by using SUSY QM and shape invariant properties from radial part of D dimensional Dirac equation and the angular quantum numbers were obtained from angular part of D dimensional Dirac equation. The SUSY operators was used to generate the D dimensional relativistic wave functions both for radial and angular parts. In the non-relativistic limit, the relativistic energy equation was reduced to the non-relativistic energy. In the classical limit, the partition function of vibrational, the specific heat of vibrational, and the mean energy of vibrational of some diatomic molecules were calculated from the equation of non-relativistic energy with the help of error function and Mat-lab 2011.

Shape invariance and SUSY separation of variables

The main ingredients of conventional Supersymmetrical Quantum Mechanics (SUSY QM) are presented. The generalization with supercharges of second order in derivatives - Second Order SUSY - is formulated, and the property of shape invariance is defined. The generalization to two-dimensional coordinate space, after using just these two elements of the modern SUSY QM approach, provides the opportunity to solve analytically some two-dimensional problems. Two different procedures of supersymmetrical separation of variables are formulated. They are illustrated by two-dimensional generalization of the Morse model.