Polynomial Eigenfunctions Research Articles

On Polynomial Eigenfunctions of a Hypergeometric-Type Operator

Consider the operator where Q(x) is some fixed polynomial of degree k. One can easily see that σQ has exactly one polynomial eigenfunction Pn(x) in each degree n ≥ 0 and its eigenvalue λn,k equals (n + k)!/h!. A more intriguing fact is that all zeros of Pn(x) lie in the convex hull of the set of zeros to Q(x). In particular, if Q(x) has only real zeros then each Pn(x) enjoys the same property. We formulate a number of conjectures on different properties of Pn(x) based on computer experiments as, for example, the interlacing property, a formula for the asymptotic distribution of zeros etc. These polynomial eigenfunctions might bethought of as a generalization of the classical Gegenbauer polynomials with half-integer superscript, this case arising when our Q(x) is an integer power of x2 – 1.

Symbolic Computation of Newton Sum Rules for the Zeros of Polynomial Eigenfunctions of Linear Differential Operators

A symbolic algorithm based on the generalized Lucas polynomials of first kind is used in order to compute the Newton sum rules for the zeros of polynomial eigenfunctions of linear differential operators with polynomial coefficients.

Quantum Bäcklund transformation for the integrable DST model

For the integrable case of the discrete self-trapping (DST) model we construct a Bäcklund transformation. The dual Lax matrix and the corresponding dual Bäcklund transformation are also found and studied. The quantum analogue of the Bäcklund transformation (Q -operator) is constructed as the trace of a monodromy matrix with an infinite-dimensional auxiliary space. We present the Q -operator as an explicit integral operator as well as describing its action on the monomial basis. As a result we obtain a family of integral equations for multivariable polynomial eigenfunctions of the quantum integrable DST model. These eigenfunctions are special functions of the Heun class which is beyond the hypergeometric class. The integral equations found are new and they shall provide a basis for efficient analytical and numerical studies of such complicated functions.

Contractions of Lie algebras: applications to special functions and separation of variables

We investigate the consequences of contraction of the Lie algebras of the orthogonal groups to the Lie algebras of the Euclidean groups in terms of separation of variables for Laplace-Beltrami eigenvalue equations, and the solutions of these equations that arise through separation of variables techniques, on the N-sphere and in N-dimensional Euclidean space. General ellipsoidal and paraboloidal coordinates are included, not just the subgroup-type coordinates that have been the concern of most investigations of contractions as applied to special functions. We pay special attention to the case N = 2 where we show in detail, for example, how Lame polynomials contract to periodic Mathieu functions. Our point of view emphasizes the characterization of separable polynomial eigenfunctions in terms of the zeros of these eigenfunctions. We also consider all possible separable coordinate systems on the complex two-sphere (which includes real hyperboloids as special cases) and their contraction to flat space coordinates.

Some properties of the Calogero - Sutherland model with reflections

We prove that the Calogero-Sutherland Model with reflections (the BC_N model) possesses a property of duality relating the eigenfunctions of two Hamiltonians with different coupling constants. We obtain a generating function for their polynomial eigenfunctions, the generalized Jacobi polynomials. The symmetry of the wave-functions for certain particular cases (associated to the root systems of the classical Lie groups B_N, C_N and D_N) is also discussed.

Generalized Hermite Polynomials and the Heat Equation for Dunkl Operators

Based on the theory of Dunkl operators, this paper presents a general concept of multivariable Hermite polynomials and Hermite functions which are associated with finite reflection groups on $\b R^N$. The definition and properties of these generalized Hermite systems extend naturally those of their classical counterparts; partial derivatives and the usual exponential kernel are here replaced by Dunkl operators and the generalized exponential kernel K of the Dunkl transform. In case of the symmetric group $S_N$, our setting includes the polynomial eigenfunctions of certain Calogero-Sutherland type operators. The second part of this paper is devoted to the heat equation associated with Dunkl's Laplacian. As in the classical case, the corresponding Cauchy problem is governed by a positive one-parameter semigroup; this is assured by a maximum principle for the generalized Laplacian. The explicit solution to the Cauchy problem involves again the kernel K, which is, on the way, proven to be nonnegative for real arguments.

Superintegrability and associated polynomial solutions: Euclidean space and the sphere in two dimensions

In this work we examine the basis functions for those classical and quantum mechanical systems in two dimensions which admit separation of variables in at least two coordinate systems. We do this for the corresponding systems defined in Euclidean space and on the two-dimensional sphere. We present all of these cases from a unified point of view. In particular, all of the special functions that arise via variable separation have their essential features expressed in terms of their zeros. The principal new results are the details of the polynomial bases for each of the nonsubgroup bases, not just the subgroup Cartesian and polar coordinate cases, and the details of the structure of the quadratic algebras. We also study the polynomial eigenfunctions in elliptic coordinates of the n-dimensional isotropic quantum oscillator.

Beam theory for strongly orthotropic materials

This paper presents a displacement-based model for orthotropic beams under plane linear elasticity based on the only kinematic assumption of transverse inextensibility. Any given axial and transverse loading as well as boundary conditions at the beam ends are considered. The solution is decomposed into the principal and the residual part (corresponding to the interior and the boundary problems) which are obtained by series expansions of polynomial functions and eigenfunctions, respectively. It is proved that the proposed one-dimensional theory gives both interior and boundary exact two-dimensional elasticity solutions for strongly orthotropic materials, i.e. for ratio between shear modulus and axial Young modulus approaching zero. For isotropic and orthotropic materials the accuracy of the beam model is also analysed and compared with that of other theories. In particular, the complementary energy error of the interior solution with respect to two-dimensional elasticity is evaluated, the asymptotic estimate of the characteristic decay length of end effects given in Choi and Horgan [J. Appl. Mech. ASME, 44, 424–430 (1977)] by two-dimensional analysis is reobtained and the range of validity of boundary solution is discussed. The numerical results presented agree very well with exact and finite element solutions even in the neighbourhood of clamped cross-sections, where the solution is mainly governed by the boundary problem.

P-adic Euler-Maclaurin expansions

An operator A of the form (Af)(x)= ∑ j a jf(cx+j), where ∑ j a j=1,0<|c|<1 , is considered in p-adic analysis. For sufficiently smooth functions f, it is shown that A m f converges to a constant function, with value say Λ( f), which in a particular case coincides with the Volkenborn integral of f. As in the real case, the asymptotic behaviour of A m f is more precisely described by a kind of Euler-Maclaurin expansion in polynomial eigenfunctions of A. This asymptotic expansion permits efficient numerical evaluation of Λ( f) by extrapolation.

The thermalized Fokker–Planck equation

The Fokker–Planck equation for a thermalized ensemble of Brownian particles is exactly solved by a method of (nontruncated) moments, using the expansion of the velocity distribution in Hermite polynomial eigenfunctions. We explicitly studied only the cases of linear and quadratic external potentials bounded by a reflecting wall. The double Laplace transforms (with respect to position and time) of the moments of the velocity distribution were obtained from the generating function, which was found to be a functional only of the density of particles. The closed equation for the density can be derived for the thermalized system. The solution of this equation (obtained by means of a Laplace transform in time) is analyzed in the long and short time limits. The (truncated) moment solution for a nonthermalized problem is also discussed. In the long time limit, the solutions for both thermalized and nonthermalized systems approach the Smoluchowski diffusive approximation.

An asymptotic expansion in wavelet analysis and its application to accurate numerical wavelet decomposition

We consider a dilation operatorT admitting a scaling function with compact support as fixed point. It is shown that the adjoint operatorT*admits a sequence of polynomial eigenfunctions and that a smooth functionf admits an expansion in these eigenfunctions, which reveals the asymptotic behavior ofT* forn??. Due to this asymptotic expansion, an extrapolation technique can be applied for the accurate numerical computation of the integrals appearing in the wavelet decomposition of a smooth function. This extrapolation technique fits well in a multiresolution scheme.

Differential Operators and the Laguerre Type Polynomials

In 1940, all fourth-order differential equations which have a sequence of orthogonal polynomial eigenfunctions were classified by H. L. Krall, up to a linear change of variable. One of these equations was subsequently named the Laguerre type equation and various properties of the orthogonal polynomial solutions and the right-definite boundary value problem were studied by A. M. Krall in 1981. In this paper, the Laguerre type expression is further studied in the right-definite setting and the appropriate left-definite problem associated with the fourth-order Laguerre type differential expression is discussed in detail.

The left-definite Legendre type boundary problem

The left-definite Legendre type boundary problem concerns the study of a fourth-order singular differential expressionMk[−] in a weighted Sobolev spaceH generated by a Dirichlet inner product. The fourth-order differential equation $$M_k [y] = \lambda y$$ has orthogonal polynomial eigenfunctions, called the Legendre type polynomials, associated with the eigenvalues $$\lambda _n = n(n + 1)(n^2 + n + 4\alpha - 2) + k.$$

AN APPLICATION OF A NEW THEOREM ON ORTHOGONAL POLYNOMIALS AND DIFFERENTIAL EQUATIONS

Abstract Recently this author proved that if an OPS satisfies a differential equation of the form then, under certain conditions, an orthogonalizing weight distribution can be found that simultaneously satisfies n distributional differential equations of orders 1,3,…2n-1. In this paper, we show how this theorem can be used to find a new sixth order equation having orthogonal polynomial eigenfunctions. The method seems more efficient than previous methods used by this author. We also discuss various properties of this OPS, including the appropriate boundary value problem.

Polynomial eigenfunctions of the exchange operator on the Fermi sphere

A complete set of eigenfunctions of the exchange operator for the (quasiparticle) scattering on the Fermi sphere is developed. The functions are simultaneously bipolynomials in the Landau variables (q/pF)2 and cos ΘL and in the Abrikosov-Khalatnikov variables cos Θ and cos Φ of the same degree. A parametrization of the quasiparticle scattering amplitude with the aid of these functions includes thes, sp, and effective potential approximations as special cases but is completely general.

On the zeros of eigenfunctions of polynomial differential operators

It is shown that for polynomial eigenfunctions of an ordinary polynomial differential operator with coefficients depending only on the independent variable it is possible to determine the density of nodes around the mean without solving the corresponding eigenvalue problem. This is done by means of the first few moments, which can be directly expressed in terms of the above-mentioned coefficients. Also, very simple expressions for the asymptotic values (i.e., when the degree of the polynomial becomes very large) of these quantities are found. For illustration, these results are applied to various orthogonal polynomials, which satisfy ordinary differential equations of second, fourth, and/or sixth order.

General polynomial expansion of the quasiparticle scattering amplitude in normal Fermi fluids

A general expansion is proposed of the quasiparticle scattering amplitude (QSA) on the Fermi sphere in polynomial eigenfunctions of the exchange operator, which approximates the QSA equally accurately in the particle-particle and both particle-hole variables. A least-squares fit is performed to the most recent sets of experimental data of the Landau parameters and transport times in the normal phase of3He. With the resulting QSA, the strong coupling corrections for the superfluid phases are calculated. The agreement with experiment is slightly less satisfactory than in previous similar attempts by other authors. This can be traced to their use of older sets of experimental data of the normal phase properties. The method is also successfully applied to3He-4He mixtures, where an upper bound toT c is given.

A modified Bars–Durgut equation with polynomial eigenfunctions

A singular integral equation in two degrees of freedom is defined. Its structure is similar to Bars–Durgut equation for baryons in two dimensions. It admits polynomial eigenfunctions and its spectrum can be studied exactly. A comparison with numerical data available for the baryon equation shows strong similarities. The asymptotic behavior of the eigenvalues for high quantum numbers is studied in the semiclassical approximation and it is found to be in good agreement with the exact spectrum. A peculiar feature of this model is the presence of a transition from a region of periodic classical orbits with constant frequency (straight Regge trajectories in the spectrum) to a regime of aperiodic orbits (nearly parabolic trajectories).

Eigenfunctions of Expected Value Operators in the Wishart Distribution

Let $(X_{1,m}, X_{2,m}, \cdots X_{k,m}), 1 \leqslant m \leqslant n,$ be a sample of size $n$ from the $k$ dimensional normal distribution with mean vector $\mu$ and covariance matrix $\Sigma.$ Let $V = (\nu_{ij}), 1 \leqslant i, j \leqslant k,$ denote the symmetric scatter matrix where $v_{ij} = \sum_m(X_{i,m} - \mu_i)(X_{j,m} - \mu_j).$ The problem posed is to characterize the eigenfunctions of the expectation operators of the Wishart distribution, i.e., those scalar valued functions, $f(V),$ such that $E(f(V)) = \lambda_{n,k}f(\Sigma).$ If $f$ is an eigenfunction then (a) for nonsingular $T,f(T'VT)$ is an eigenfunction and (b) for integral $p, |V|^{p/2}f(V)$ is an eigenfunction. For $k \leqslant 2,$ a complete solution of the problem is given. For $k = 1$ the functions $f(v) = cv^\alpha$ are the only eigenfunctions. For $k = 2,$ a function $f$ is an eigenfunction if and only if (i) $f$ is homogeneous and (ii) $4 \frac{\partial^2 f}{\partial v_{22}\partial v_{11}} - \frac{\partial^2 f}{\partial^2v_{21}} = C|V|^{-1}f.$ A representation of eigenfunctions is given in terms of sums of associated Legendre functions. Relationships between eigenfunctions and harmonic functions are indicated. Any homogeneous polynomial is proved to be a linear combination of polynomial eigenfunctions.

Erratum to: Singular integral operators with integral eigenvalues and polynomial eigenfunctions

Erratum to: Singular integral operators with integral eigenvalues and polynomial eigenfunctions