Construction of solvable quantum mechanical potentials

In recent years, one of the most interesting developments in quantum mechanics has been the construction of new exactly solvable potentials connected with the appearance of families of exceptional orthogonal polynomials (EOP) in mathematical physics. In contrast with families of (Jacobi, Laguerre and Hermite) classical orthogonal polynomials, which start with a constant, the EOP families begin with some polynomial of degree greater than or equal to one, but still form complete, orthogonal sets with respect to some positive-definite measure. We show how they may appear in the bound-state wavefunctions of some rational extensions of well-known exactly solvable quantum potentials. Such rational extensions are most easily constructed in the framework of supersymmetric quantum mechanics (SUSYQM), where they give rise to a new class of translationally shape invariant potentials. We review the most recent results in this field, which use higher-order SUSYQM. We also comment on some recent re-examinations of the shape invariance condition, which are independent of the EOP construction problem.

In this work, a canonical point and a gauge transformation applied to theg(x) times differential equations (DE) for special functions (SF) is used to convert it into a Schrödinger‐like equation. This method leads to exactly solvable potentials allowing special functions solutions. For specific applications, keeping in mind the general DE for SF, the choice of the coefficients that identify the SF under study allows to find the equivalent of the Witten superpotential and wavefunctions for eachg(x). That is, the choice of a particularg(x) leads straightforwardly to those exactly solvable potentials having as wavefunctions the SF under consideration. As a useful application of the proposed method, we considered explicitly the potentials associated to the standard Hermite, Laguerre, Legendre, and Bessel SF in the case of some particular forms ofg(x). However, the proposed approach can be directly applied to any other standard SF with the aim to obtain exactly solvable potentials that can be used in the modelling of quantum chemical applications. © 2008 Wiley Periodicals, Inc. Int J Quantum Chem, 2008

We present a unified approach for solving and classifying exactly solvable\npotentials. Our unified approach encompasses many well-known exactly solvable\npotentials. Moreover, the new approach can be used to search systematically for\na new class of solvable potentials.\n

We consider the eigenvalue problem associated with the Dunkl-type differential operator (in which the reflection operator R is involved)in the context of supersymmetric quantum mechanical models. By solving this eigenvalue problem with the help of known exactly solvable potentials, we construct several classes of functions satisfying certain orthogonality relations. We call them the Dunkl-supersymmetric (Dunkl-SUSY) orthogonal functions. These functions can be expressed in terms of the classical orthogonal polynomials. The key feature of these functions is that they appear by pairs, i.e. Qn(x) and Qn(−x) are both the eigenfunctions of . A general formulation of the Dunkl-SUSY orthogonal polynomials is also presented.

Minimal recurrence relations for connection coefficients between classical orthogonal polynomials: Continuous case

Quadrature formulas for Bessel polynomials

For all quantum-mechanical potentials that are known to be exactly solvable, there are two different, and seemingly independent methods of solution. The first approach is the potential algebra of symmetry groups; the second is supersymmetric quantum mechanics, applied to shape-invariant potentials, which comprise the set of known exactly solvable potentials. Using the underlying algebraic structures of Natanzon potentials, of which the translational shape-invariant potentials are a special subset, we demonstrate the equivalence of the two methods of solution. In addition, we show that, while the algebra for the general Natanzon potential is so(2,2), the subgroup so(2,1) suffices for the shape invariant subset. Finally, we show that the known set of exactly solvable potentials in fact constitutes the full set of such potentials.

The location of the zeros of general hypergeometric polynomials are linked with those of the classical orthogonal polynomials in some cases, notably 2 F 1 and 1 F 1 hypergeometric polynomials which have been extensively studied. In the case of 3 F 2 polynomials, less is known about their properties, including the location of their zeros, because there is, in general, no direct link with orthogonal polynomials. We consider two classes of 3 F 2 hypergeometric polynomials, each of which has a representation in terms of 2 F 1 polynomials. Our main result proves that the class of polynomials 3 F 2(– n, a, b;a – 1, d; x), a, b, d ∈ R, n ∈ N is quasi-orthogonal of order 1 on an interval that varies with the real parameters b and d. We discuss the apparent role played by the parameter a with regard to the location of all the zeros of this class of polynomials. We also prove results on the location of the zeros of the class 3 F 2(– n, b,(b – n)/2; b – n, (b – n – 1)/2; x), b ∈ R, n ∈N by using the orthogonality and quasi-orthogonality of factors involved in its representation.

Connection and linearization coefficients of the Askey–Wilson polynomials

Fourth-order difference equation for the associated classical discrete orthogonal polynomials

‎In this article‎, ‎an efficient numerical method based on a new class of orthogonal polynomials‎, ‎namely Chelyshkov polynomials‎, ‎has been presented to approximate solution of time-fractional telegraph (TFT) equations‎. ‎The fractional operational matrix of the Chelyshkov polynomials along with the typical collocation method is used to reduces TFT equations to a system of algebraic equations‎. ‎The error analysis of the proposed collocation method is also investigated‎. ‎A comparison with other published results confirms that the presented Chelyshkov collocation approach is efficient and accurate for solving TFT equations‎. ‎Illustrative examples are included to demonstrate the efficiency of the Chelyshkov method‎.

In this paper, we introduce the concept of the \(\mathbb{B}_{\alpha}\)-classical orthogonal polynomials, where \(\mathbb{B}_{\alpha}\) is the raising operator \(\mathbb{B}_{\alpha}:=x^2 \cdot {d}/{dx}+\big(2(\alpha-1)x+1\big)\mathbb{I}\), with nonzero complex number \(\alpha\) and \(\mathbb{I}\) representing the identity operator. We show that the Bessel polynomials \(B^{(\alpha)}_n(x),\ n\geq0\), where \(\alpha\neq-{m}/{2}, \ m\geq -2, \ m\in \mathbb{Z}\), are the only \(\mathbb{B}_{\alpha}\)-classical orthogonal polynomials. As an application, we present some new formulas for polynomial solution.

We investigate approximations of functions of classes W2(Dγ;(a,b)), r = 2, 3, …, by classical orthogonal polynomials with a weight γ in the spaces L2,γ(a,b). We obtain upper and lower estimates for different widths on the classes W2(Ωm,γ, Ψ; (a,b)), where r ∈ ℤ+, m ∈ ℕ, Ψ is a majorant, Ωm,γ is a generalized modulus of continuity of m-th order. We find the condition on majorant, which enable us to compute the exact values of widths, and give certain examples of these values. In all mentioned above classes we obtain bounds (including the least upper bounds) for the Fourier coefficients.

A transformation method was applied to the biconfluent (BHE), doubly confluent (DHE), and triconfluent (THE) Heun equations to generate and classify exactly solvable quantum mechanical potentials derived from them. With this, the range of potentials solvable in terms of the confluent hypergeometric function can be extended. The resulting potentials contained five independently tunable terms and two terms originating from the Schwartzian derivative that depended only on the parameters of the z(x) transformation function. The polynomial solutions of these potentials contain expansion coefficients obtained from three-term (BHE and DHE) and four-term (THE) recurrence relations. For the simplest z(x) transformation functions, the Lemieux–Bose potentials have been recovered for the BHE and DHE. The coupling parameters of these potentials and also of five potentials derived from the THE have been expressed in terms of the parameters of the respective differential equations. The present scheme offers a general framework into which a number of earlier results can be integrated in a systematic way. These include special cases of potentials obtained from less general versions of the Heun-type equations and individual solvable potentials obtained from various methods that do not necessarily refer to the Heun-type equations considered here. Several potentials derived here were found to coincide with or reduce to potentials found earlier from the quasi-exactly solvable (QES) formalism. Based on their mathematical form, their physically relevant features (domain of definition, asymptotic behaviour, single- or multi-well structure) were discussed, and possible fields of applications were pointed out.

The zeros and asymptotic limits of two new classes of orthogonal polynomials, which are derived by applying two orthogonalization procedures due to Lowdin to a set of monomials, are calculated. It is established that they possess all the properties ofthe zeros of a polynomial. Their asymptotic limits are found. A Unified view of all the Lowdin orthogonal polynomials together with the standard classical orthogonal polynomials are presented in a unique graph.