A new type of exact solvability is reported. We study the general central polynomial potentials (with 2q anharmonic terms) which satisfy the Magyari's partial exact solvability conditions (this means that they possess a harmonic-oscillator-like wave function proportional to a polynomial of any integer degree N). Working in the space of a very large dimension D for simplicity, we reveal that in contrast to the usual version of the model in finite dimensions (requiring a purely numerical treatment of the Magyari's constraints), our large D problem acquires an explicit, closed form solution at all N and up to q = 5 at least. This means that our effective secular polynomials (generated via the standard technique of Groebner bases) happen to be all fully factorizable in an utterly mysterious manner (mostly, over integers).
The aim of this paper is to study generalized q-analogs of the well-known q-deformed harmonic oscillators and to connect them with q-Hermite polynomials. We give a construction of the appropriate oscillator-like algebras and show that corresponding Hermite polynomials are generalization of the discrete q-Hermite I and the discrete q-Hermite II polynomials. We also construct generalized coherent states of Barut-Girardello type for oscillator-like systems connected with these polynomials.
The polynomial algebra is a deformed SU(2) algebra. Here, we use polynomial algebra as a method to solve a series of deformed oscillators. Meanwhile, we find a series of physics systems corresponding with polynomial algebra with different maximal order.
This work addresses a full characterization of three new q-polynomials derived from the $q-$oscillator algebra. Related matrix elements and generating functions are deduced. Further, a connection between Hahn factorial and q-Gaussian polynomials is established.
A construction is given of the most general representations of the q-oscillator algebra where both generators are tridiagonal. It is shown to be connected to the Askey-Wilson polynomials.
We discuss the construction of oscillator-like systems associated with orthogonal polynomials on the example of the Fibonacci oscillator. In addition, we consider the dimension of the corresponding lie algebras.
A set of r non-Hermitian oscillator Hamiltonians in r dimensions is shown to be simultaneously diagonalizable. Their spectra is real and the common eigenstates are expressed in terms of multiple Charlier polynomials. An algebraic interpretation of these polynomials is thus achieved and the model is used to derive some of their properties.
In this paper we first construct an analytic realization of the $C_λ$-extended oscillator algebra with the help of difference-differential operators. Secondly, we study families of $d$-orthogonal polynomials which are extensions of the Hermite and Laguerre polynomials. The underlying algebraic framework allowed us a systematic derivation of their main properties such as recurrence relations, difference-differential equations, lowering and rising operators and generating functions. Finally, we use these polynomials to construct a realization of the $C_λ$-extended oscillator by block matrices.
The factorization method of Infeld and Hull is applied to the radial Schrödinger equation for $d$-dimensional isotropic harmonic oscillator and various ladder operators are defined. The radial energy eigenstates are expressed in terms of the generalized Laguerre polynomials and their properties are shown to follow from the expressions involving the ladder operators. In the same way as the harmonic oscillator we also obtain the bound energy eigenstates of the Morse oscillator.
In this paper, we present an explicit realization of q-deformed Calogero-Vasiliev algebra whose generators are first-order q-difference operators related to the generalized discrete q-Hermite II polynomials recently introduced in [13]. Furthermore, we construct the wave functions and we determine the q-coherent states.
The asymptotics of the weighted $L_{p}$-norms of Hermite polynomials, which describes the Rényi entropy of order $p$ of the associated quantum oscillator probability density, is determined for $n\to\infty$ and $p>0$. Then, it is applied to the calculation of the Rényi entropy of the quantum-mechanical probability density of the highly-excited (Rydberg) states of the isotropic oscillator.
We present the most general polynomial Lie algebra generated by a second order integral of motion and one of order M, construct the Casimir operator, and show how the Jacobi identity provides the existence of a realization in terms of deformed oscillator algebra. We also present the classical analog of this construction for the most general Polynomial Poisson algebra. Two specific classes of such polynomial algebras are discussed that include the symmetry algebras observed for various 2D superintegrable systems.
We revisit the radial oscillator from the free oscillator realization point of view. By using a free oscillator, namely the creation/annihilation operators of the harmonic oscillator, we construct an operator that maps the eigenfunctions of the harmonic oscillator to those of the radial oscillator. As a polynomial part of this relation, we obtain an operator that maps the Hermite polynomials to the Laguerre polynomials.
Karabulut and Sibert (\textit{J. Math. Phys}. \textbf{38} (9), 4815 (1997)) have constructed an orthogonal set of functions from linear combinations of equally spaced Gaussians. In this paper we show that they are actually eigenfunctions of a q-oscillator in coordinate representation. We also reinterpret the coordinate representation example of q-oscillator given by Macfarlane as the functions orthogonal with respect to an unusual inner product definition. It is shown that the eigenfunctions in both q-oscillator examples are infinitely degenerate.
There is a generalized oscillator-like algebra associated with every class of orthogonal polynomials $\{Ψ_n(x)\}_{n=0}^{\infty}$, on the real line, satisfying a four term non-symmetric recurrence relation $xΨ_n(x)=b_nΨ_{n+1}(x)+a_nΨ_n(x)+b_{n-1}Ψ_{n-1}(x),~Ψ_0(x)=1,~b_{-1}=0$. This note presents necessary and sufficient conditions on $a_n$ and $b_n$ for such algebras to be of finite dimension. As examples, we discuss the dimensions of oscillator-like algebras associated with Laguerre and Jacobi polynomials.
A simple discrete model of the two dimensional isotropic harmonic oscillator is presented. It is superintegrable with su(2) as its symmetry algebra. It is constructed with the help of the algebraic properties of the bivariate Charlier polyno-mials. This adds to the other discrete superintegrable models of the oscillator based on Krawtchouk and Meixner orthogonal polynomials in several variables.
We modify a non-perturbative approach based on the Principle of Minimal Sensitivity to calculate the periods of relativistic oscillators with even polynomial potentials of general form. The optimization of the variational parameter is adapted by using the ultrarelativistic limit of the period as a boundary condition, which determines the values of additional free parameters introduced. Resulting general approximations prove to be accurate over the whole solution domain and for the whole class of potentials.
The multivariate Meixner polynomials are shown to arise as matrix elements of unitary representations of the $SO(d,1)$ group on oscillator states. These polynomials depend on $d$ discrete variables and are orthogonal with respect to the negative multinomial distribution. The emphasis is put on the bivariate case for which the SO(2,1) connection is used to derive the main properties of the polynomials: orthogonality relation, raising/lowering relations, generating function, recurrence relations and difference equations as well as explicit expressions in terms of standard (univariate) Krawtchouk…
We prove a reducibility result for a quantum harmonic oscillator in arbitrary dimensions with arbitrary frequencies perturbed by a linear operator which is a polynomial of degree two in $x_j$, $-i \partial_j$ with coefficients which depend quasiperiodically on time.
An algebraic interpretation of the bivariate Krawtchouk polynomials is provided in the framework of the 3-dimensional isotropic harmonic oscillator model. These polynomials in two discrete variables are shown to arise as matrix elements of unitary reducible representations of the rotation group in 3 dimensions. Many of their properties are derived by exploiting the group-theoretic setting. The bivariate Tratnik polynomials of Krawtchouk type are seen to be special cases of the general polynomials that correspond to particular rotations involving only two parameters. It is explained how the app…
An explicit expression for the general bivariate Krawtchouk polynomials is obtained in terms of the standard Krawtchouk and dual Hahn polynomials. The bivariate Krawtchouk polynomials occur as matrix elements of the unitary reducible representations of SO(3) on the energy eigenspaces of the 3-dimensional isotropic harmonic oscillator and the explicit formula is obtained from the decomposition of these representations into their irreducible components. The decomposition entails expanding the Cartesian basis states in the spherical bases that span irreducible SO(3) representations. The overlap c…
A family of multivariate orthogonal polynomials generalizing the standard (univariate) Charlier polynomials is shown to arise in the matrix elements of the unitary representation of the Euclidean group E(d) on oscillator states. These polynomials in d discrete variables are orthogonal on the product of d Poisson distributions. The accent is put on the d=2 case and the group theoretical setting is used to obtain the main properties of the polynomials: orthogonality and recurrence relations, difference equation, raising/lowering relations, generating function, hypergeometric and integral represe…
In this paper, we calculate the Mandel parameter $Q_M$ for an oscillator-like system generated by generalized Chebyshev polynomials \cite{01}, \cite{02}, \cite{03}. The sign of the Mandel parameter $Q_M$ characterizes the deviation of the excitation statistics from the Poisson one. This work is a continuation of our works \cite{04}, \cite{05}.
It is shown that fundamental solutions $K^σ(x,y;t)=\langle x|e^{-i H^σt}|y\rangle$ of the non-stationary Schrödinger equation (Green functions, or propagators) for the rational extensions of the Harmonic oscillator $H^σ=H_{\rm osc}+ΔV^σ$ are expressed in terms of elementary functions only. An algorithm to calculate explicitly $K_σ$ for an arbitrary $σ\in \mathbb{N}$ is given, compact expressions for $K^{\{1,2\}}$ and $K^{\{2,3\}}$ are presented. A generalization of the Mehler's formula to the case of exceptional Hermite polynomials is given.
A set of generalized squeezed-coherent states for the finite u(2) oscillator is obtained. These states are given as linear combinations of the mode eigenstates with amplitudes determined by matrix elements of exponentials in the su(2) generators. These matrix elements are given in the (N+1)-dimensional basis of the finite oscillator eigenstates and are seen to involve 3x3 matrix multi-orthogonal polynomials Q_n(k) in a discrete variable k which have the Krawtchouk and vector-orthogonal polynomials as their building blocks. The algebraic setting allows for the characterization of these polynomi…
There is a generalized oscillator algebra associated with every class of orthogonal polynomials $\{Ψ_n(x)\}_{n=0}^{\infty}$, on the real line, satisfying a three term recurrence relation $xΨ_n(x)=b_nΨ_{n+1}(x)+b_{n-1}Ψ_{n-1}(x), Ψ_0(x)=1, b_{-1}=0$. This note presents necessary and safficient conditions on $b_n$ for such algebras to be of finite dimension. As examples, we discuss the dimensions of oscillator algebras associated with Hermite, Legendre and Gegenbauer polynomials. In addition we shall also discuss the dimensions of some generalized deformed oscillator algebras. Some remarks on th…
We develop a variational method to obtain accurate bounds for the eigenenergies of H = -Delta + V in arbitrary dimensions N>1, where V(r) is the nonpolynomial oscillator potential V(r) = r^2 + lambda r^2/(1+gr^2), lambda in (-infinity,\infinity), g>0. The variational bounds are compared with results previously obtained in the literature. An infinite set of exact solutions is also obtained and used as a source of comparison eigenvalues.
A (p,q)-analogue of the classical Rogers-Szego polynomial is defined by replacing the q-binomial coefficient in it by the (p,q)-binomial coefficient. Exactly like the Rogers-Szego polynomial is associated with the q-oscillator algebra it is found that the (p,q)-Rogers-Szego polynomial is associated with the (p,q)-oscillator algebra.
Under certain constraints on the parameters a, b and c, it is known that Schroedinger's equation -y"(x)+(ax^6+bx^4+cx^2)y(x) = E y(x), a > 0, with the sextic anharmonic oscillator potential is exactly solvable. In this article we show that the exact wave function y is the generating function for a set of orthogonal polynomials P_n^{(t)}(x) in the energy variable E. Some of the properties of these polynomials are discussed in detail and our analysis reveals scaling and factorization properties that are central to quasi-exact solvability. We also prove that this set of orthogonal polynomials can…
The symmetrized quartic polynomial oscillator is shown to admit an sl(2,$\R$) algebraization. Some simple quasi-exactly solvable (QES) solutions are exhibited. A new symmetrized sextic polynomial oscillator is introduced and proved to be QES by explicitly deriving some exact, closed-form solutions by resorting to the functional Bethe ansatz method. Such polynomial oscillators include two categories of QES potentials: the first one containing the well-known analytic sextic potentials as a subset, and the second one of novel potentials with no counterpart in such a class.
In the interesting paper G. Honnouvo and K. Thirulogasanthar [J. Math. Phys. {\bf 55} , 093511 (2014)] the authors obtained the necessary and sufficient conditions under which the oscillator algebra connected with orthogonal polynomials on real line is finite-dimensional (and in this case the dimension of the algebra is always equal four). In the cited article, only the case when polynomials are orthogonal with respect to a symmetric measure on the real axis was considered. Unfortunately, the sufficient condition from this paper is incomplete. Here we clarify the sufficient part of the corresp…
A remarkable extension of Rayleigh-Schroedinger perturbation method is found. Its (N+q) x (N+1) - dimensional Hamiltonians (as emerging, e.g., during quasi-exact constructions of bound states) are non-square matrices at q > 1. The role of an eigenvalue is played by an energy/coupling q-plet. In all orders, its perturbations are defined via a q-dimensional inversion.
We construct a new class of coherent states labeled by points z of the complex plane and depending on three numbers (gamma, nu) and epsilon positive by replacing the coefficients of the canonical coherent states by Laguerre polynomials. These states are superpositions of eigenstates of the symmetric Poschl-Teller oscillator and they solve the identity of the states Hilbert space at the limit epsilon goes to 0. Their wavefunctions are obtained in a closed form for a special case of parameters (gamma, nu). We discuss their associated coherent states transform which leads to an integral represent…
We construct a class of generalized phase coherent states indexed by points of the unit circle and depending on three positive parameters "gamma","alpha" and "epsilon" by replacing the labelling coefficients of the canonical coherent states by circular Jacobi polynomials with parameter "gamma". The special case "gamma" = 0 corresponds to well known phase coherent phase states. The constructed states are superposition of eigenstates of a one-parameter pseudoharmonic oscillator depending on "alpha" and solve the identity of the state Hilbert space at the limit "epsilon"->0+. Closed form for thei…
We consider two families of polynomials $\mathbb{P}=\polP$ and $\mathbb{Q}=\polQ$\footnote{Here and below we consider only monic polynomials.} orthogonal on the real line with respect to probability measures $μ$ and $ν$ respectively. Let $\polQ$ and $\polP$ connected by the linear relations $$ Q_n(x)=P_n(x)+a_1P_{n-1}(x)+...+a_kP_{n-k}(x).$$ Let us denote $\mathfrak{A}_P$ and $\mathfrak{A}_Q$ generalized oscillator algebras associated with the sequences $\mathbb{P}$ and $\mathbb{Q}$. In the case $k=2$ we describe all pairs ($\mathbb{P}$,$\mathbb{Q}$), for which the algebras $\mathfrak{A}_P$ an…
The oscillator Racah algebra $\mathcal{R}_n(\mathfrak{h})$ is realized by the intermediate Casimir operators arising in the multifold tensor product of the oscillator algebra $\mathfrak{h}$. An embedding of the Lie algebra $\mathfrak{sl}_{n-1}$ into $\mathcal{R}_n(\mathfrak{h})$ is presented. It relates the representation theory of the two algebras. We establish the connection between recoupling coefficients for $\mathfrak{h}$ and matrix elements of $\mathfrak{sl}_n$-representations which are both expressed in terms of multivariate Krawtchouk polynomials of Griffiths type.
Karlin and McGregor's d-variable Hahn polynomials are shown to arise in the (d+1)-dimensional singular oscillator model as the overlap coefficients between bases associated to the separation of variables in Cartesian and hyperspherical coordinates. These polynomials in d discrete variables depend on d+1 real parameters and are orthogonal with respect to the multidimensional hypergeometric distribution. The focus is put on the d=2 case for which the connection with the three-dimensional singular oscillator is used to derive the main properties of the polynomials: forward/backward shift operator…
In this paper, by searching the relation between the radial part of Higgs harmonic oscillator in the two-dimensional curved space and the generalized CRS harmonic oscillator model, we can find a series of quasi-exact models in two-dimensional curved space based on this relation.
A previous study of exactly solvable rationally-extended radial oscillator potentials and corresponding Laguerre exceptional orthogonal polynomials carried out in second-order supersymmetric quantum mechanics is extended to $k$th-order one. The polynomial appearing in the potential denominator and its degree are determined. The first-order differential relations allowing one to obtain the associated exceptional orthogonal polynomials from those arising in a ($k-1$)th-order analysis are established. Some nontrivial identities connecting products of Laguerre polynomials are derived from shape in…
In this paper we present a straightforward systematic method for the exact and approximate calculation of integrals that appear in formulas for the period of anharmonic oscillators and other problems of interest in classical mechanics.
Multiple Meixner polynomials are polynomials in one variable which satisfy orthogonality relations with respect to $r>1$ different negative binomial distributions (Pascal distributions). There are two kinds of multiple Meixner polynomials, depending on the selection of the parameters in the negative binomial distribution. We recall their definition and some formulas and give generating functions and explicit expressions for the coefficients in the nearest neighbor recurrence relation. Following a recent construction of Miki, Tsujimoto, Vinet and Zhedanov (for multiple Meixner polynomials of th…
A superintegrable, discrete model of the quantum isotropic oscillator in two-dimensions is introduced. The system is defined on the regular, infinite-dimensional $\mathbb{N}\times \mathbb{N}$ lattice. It is governed by a Hamiltonian expressed as a seven-point difference operator involving three parameters. The exact solutions of the model are given in terms of the two-variable Meixner polynomials orthogonal with respect to the negative trinomial distribution. The constants of motion of the system are constructed using the raising and lowering operators for these polynomials. They are shown to …
We consider a one-parameter family of nonlinear coherent states by replacing the factorial in coefficients of the canonical coherent states by a specific generalized factorial depending on a parameter gamma. These states are superposition of eigenstates of the Hamiltonian with a symmetric Poschl-Teller potential depending on a parameter nu > 1. The associated Bargmann-type transform is defined for equal parameters. Some results on the infinite square well potential are also derived. For some different values of gamma, we discuss two sets of orthogonal polynomials that are naturally attached to…
We show that the crossing symmetric dispersion relation (CSDR) for 2-2 scattering leads to a fascinating connection with knot polynomials and q-deformed algebras. In particular, the dispersive kernel can be identified naturally in terms of the generating function for the Alexander polynomials corresponding to the torus knot $(2,2n+1)$ arising in knot theory. Certain linear combinations of the low energy expansion coefficients of the amplitude can be bounded in terms of knot invariants. Pion S-matrix bootstrap data respects the analytic bounds so obtained. We correlate the $q$-deformed harmonic…
A family of multi-parameter, polynomially deformed oscillators (PDOs) given by polynomial structure function ϕ(n) is studied from the viewpoint of being (or not) in the class of Fibonacci oscillators. These obey the Fibonacci relation/property (FR/FP) meaning that the n-th level energy E_n is given linearly, with real coefficients, by the two preceding ones E_{n-1}, E_{n-2}. We first prove that the PDOs do not fall in the Fibonacci class. Then, three different paths of generalizing the usual FP are developed for these oscillators: we prove that the PDOs satisfy respective k-term generalized Fi…
By factorization of the Hamiltonian describing the quantum mechanics of the continuous q-Hermite polynomial, the creation and annihilation operators of the q-oscillator are obtained. They satisfy a q-oscillator algebra as a consequence of the shape-invariance of the Hamiltonian. A second set of q-oscillator is derived from the exact Heisenberg operator solution. Now the q-oscillator stands on the equal footing to the ordinary harmonic oscillator.
The present letter obtains the exact solution and geometric phase of the time-dependent Schrödinger equation governing the dipole oscillator in the exterior electric field, by making use of the Lewis-Riesenfeld invariant theory and the invariant-related unitary transformation formulation. It is shown that the geometric phase presents the global property of the time evolution of the dipole oscillator in time-dependent environments.
We transform the time-dependent Schroedinger equation for the most general variable quadratic Hamiltonians into a standard autonomous form. As a result, the time-evolution of exact wave functions of generalized harmonic oscillators is determined in terms of solutions of certain Ermakov and Riccati-type systems. In addition, we show that the classical Arnold transformation is naturally connected with Ehrenfest's theorem for generalized harmonic oscillators.
The type III Hermite $X_m$ exceptional orthogonal polynomial family is generalized to a double-indexed one $X_{m_1,m_2}$ (with $m_1$ even and $m_2$ odd such that $m_2 > m_1$) and the corresponding rational extensions of the harmonic oscillator are constructed by using second-order supersymmetric quantum mechanics. The new polynomials are proved to be expressible in terms of mixed products of Hermite and pseudo-Hermite ones, while some of the associated potentials are linked with rational solutions of the Painlevé IV equation. A novel set of ladder operators for the extended oscillators is also…
In Oste and Van der Jeugt, SIGMA, 12 (2016) we classified all pairs of recurrence relations in which two (dual) Hahn polynomials with different parameters appear. Such pairs are referred to as (dual) Hahn doubles, and the same technique was then applied to obtain all Racah doubles. We now consider a special case concerning the doubles related to Racah polynomials. This gives rise to an interesting class of two-diagonal matrices with closed form expressions for the eigenvalues. Just as it was the case for (dual) Hahn doubles, the resulting two-diagonal matrix can be used to construct a finite o…
We prove that every rational extension of the quantum harmonic oscillator that is exactly solvable by polynomials is monodromy free, and therefore can be obtained by applying a finite number of state-deleting Darboux transformations on the harmonic oscillator. Equivalently, every exceptional orthogonal polynomial system of Hermite type can be obtained by applying a Darboux-Crum transformation to the classical Hermite polynomials. Exceptional Hermite polynomial systems only exist for even codimension 2m, and they are indexed by the partitions λof m. We provide explicit expressions for their cor…
The two-dimensional extension of the one-dimensional PDM-Lagrangians and their nonlocal point transformation mappings into constant unit-mass exactly solvable Lagrangians is introduced. The conditions on the related two-dimensional Euler-Lagrange equations' invariance are reported. The mappings from superintegrable linear oscillators into sub-superintegrable nonlinear PDM-oscillators are exemplified by, (i) a sub-superintegrable Mathews-Lakshmanan type-I PDM-oscillator, for a PDM-particle moving in a harmonic oscillator potential, (ii) a sub-superintegrable Mathews-Lakshmanan type-II PDM-oscil…
Extending the supersymmetric method proposed by Tkachuk to the complex domain, we obtain general expressions for superpotentials allowing generation of quasi-exactly solvable PT-symmetric potentials with two known real eigenvalues (the ground state and first-excited state energies). We construct examples, namely those of complexified non-polynomial oscillators and of a complexified hyperbolic potential, to demonstrate how our scheme works in practice. For the former we provide a connection with the sl(2) method, illustrating the comparative advantages of the supersymmetric one.
We study the eigenvalue problem $-u^{\prime\prime}(z)-[(iz)^m+P_{m-1}(iz)]u(z)=λu(z)$ with the boundary conditions that $u(z)$ decays to zero as $z$ tends to infinity along the rays $\arg z=-\fracπ{2}\pm \frac{2π}{m+2}$, where $P_{m-1}(z)=a_1 z^{m-1}+a_2 z^{m-2}+...+a_{m-1} z$ is a polynomial and integers $m\geq 3$. We provide an asymptotic expansion of the eigenvalues $λ_n$ as $n\to+\infty$, and prove that for each {\it real} polynomial $P_{m-1}$, the eigenvalues are all real and positive, with only finitely many exceptions.
We present the quadratic algebra of the generalized MICZ-Kepler system in three-dimensional Euclidean space $E_{3}$ and its dual the four dimensional singular oscillator in four-dimensional Euclidean space $E_{4}$. We present their realization in terms of a deformed oscillator algebra using the Daskaloyannis construction. The structure constants are in these cases function not only of the Hamiltonian but also of other integrals commuting with all generators of the quadratic algebra. We also present a new algebraic derivation of the energy spectrum of the MICZ-Kepler system on the three sphere …
We investigate an algebraic model for the quantum oscillator based upon the Lie superalgebra sh(2|2), known as the Heisenberg-Weyl superalgebra or "the algebra of supersymmetric quantum mechanics", and its Fock representation. The model offers some freedom in the choice of a position and a momentum operator, leading to a free model parameter gamma. Using the technique of Jacobi matrices, we determine the spectrum of the position operator, and show that its wavefunctions are related to Charlier polynomials C_n with parameter gamma^2. Some properties of these wavefunctions are discussed, as well…
