An interesting discovery in the last two years in the field of mathematical physics has been the exceptional $X_\ell$ Laguerre and Jacobi polynomials. Unlike the well-known classical orthogonal polynomials which start with constant terms, these new polynomials have the lowest degree $\ell=1,2,...$, and yet they form complete sets with respect to some positive-definite measure. In this paper, we study one important aspect of these new polynomials, namely, the behaviors of their zeros as some parameters of the Hamiltonians change.
The multi-indexed Laguerre and Jacobi polynomials form a complete set of orthogonal polynomials. They satisfy second-order differential equations but not three term recurrence relations, because of the 'holes' in their degrees. The multi-indexed Laguerre and Jacobi polynomials have Wronskian expressions originating from multiple Darboux transformations. For the ease of applications, two different forms of simplified expressions of the multi-indexed Laguerre and Jacobi polynomials are derived based on various identities. The parity transformation property of the multi-indexed Jacobi polynomials…
In a previous paper we derived equivalence relations for pseudo-Wronskian determinants of Hermite polynomials. In this paper we obtain the analogous result for Laguerre and Jacobi polynomials. The equivalence formulas are richer in this case since rational Darboux transformations can be defined for four families of seed functions, as opposed to only two families in the Hermite case. The pseudo-Wronskian determinants of Laguerre and Jacobi type will thus depend on two Maya diagrams, while Hermite pseudo-Wronskians depend on just one Maya diagram. We show that these equivalence relations can be …
A method to construct multi-indexed exceptional Laguerre polynomials using isospectral deformation technique and quantum Hamilton-Jacobi (QHJ) formalism is presented. We construct generalized superpotentials using singularity structure analysis, which lead to rational potentials with multi-indexed polynomials as solutions. We explicitly construct such rational extensions of the radial oscillator and their solutions, which involve exceptional Laguerre orthogonal polynomials having two indices. The exact expressions for the $L1$, $L2$ and $L3$ type polynomials, along with their weight functions …
A New Class of Exceptional Orthogonal Polynomials: The Type III $X_{m}$-Laguerre Polynomials And The Spectral Analysis of Three Types of Exceptional Laguerre Polynomials
The present paper is about Bernstein-type estimates for Jacobi polynomials and their applications to various branches in mathematics. This is an old topic but we want to add a new wrinkle by establishing some intriguing connections with dispersive estimates for a certain class of Schrödinger equations whose Hamiltonian is given by the generalized Laguerre operator. More precisely, we show that dispersive estimates for the Schrödinger equation associated with the generalized Laguerre operator are connected with Bernstein-type inequalities for Jacobi polynomials. We use known uniform estimates f…
As the fourth stage of the project multi-indexed orthogonal polynomials, we present the multi-indexed Meixner and little $q$-Jacobi (Laguerre) polynomials in the framework of `discrete quantum mechanics' with real shifts defined on the semi-infinite lattice in one dimension. They are obtained, in a similar way to the multi-indexed Laguerre and Jacobi polynomials reported earlier, from the quantum mechanical systems corresponding to the original orthogonal polynomials by multiple application of the discrete analogue of the Darboux transformations or the Crum-Krein-Adler deletion of virtual stat…
We introduce two ordinary second-order linear differential equations of the Laguerre- and Jacobi-type. Solutions are written as infinite series of square integrable functions in terms of the Laguerre and Jacobi polynomials, respectively. The expansion coefficients of the series satisfy three-term recursion relations, which are solved in terms of orthogonal polynomials with continuous and/or discrete spectra. Most of these are well-known polynomials whereas few are not. We present physical applications of these differential equations in quantum mechanics.
We define Laguerre and Jacobi analogues of the Warren process. That is, we construct local dynamics on a triangular array of particles so that the projections to each level recover the Laguerre and Jacobi eigenvalue processes of König-O'Connell and Doumerc and the fixed time distributions recover the joint distribution of eigenvalues in multilevel Laguerre and Jacobi random matrix ensembles. Our techniques extend and generalize the framework of intertwining diffusions developed by Pal-Shkolnikov. One consequence is the construction of particle systems with local interactions whose fixed time d…
We obtain large $n$ asymptotics of $n \times n$ Hankel determinants whose weight has a one-cut regular potential and Fisher-Hartwig singularities. We restrict our attention to the case where the associated equilibrium measure possesses either one soft edge and one hard edge (Laguerre-type) or two hard edges (Jacobi-type). We also present some applications in the theory of random matrices. In particular, we can deduce from our results asymptotics for partition functions with singularities, central limit theorems, correlations of the characteristic polynomials, and gap probabilities for (piecewi…
Under the Kolmogorov--Smirnov metric, an upper bound on the rate of convergence to the Gaussian distribution is obtained for linear statistics of the matrix ensembles in the case of the Gaussian, Laguerre, and Jacobi weights. The main lemma gives an estimate for the characteristic functions of the linear statistics; this estimate is uniform over the growing interval. The proof of the lemma relies on the Riemann--Hilbert approach.
We show that for $β\ge 1$ the semigroups of $β$-Laguerre and $β$-Jacobi processes of different dimensions are intertwined in analogy to a similar result for $β$-Dyson Brownian motion recently obtained by Ramanan and Shkolnikov. These intertwining relations generalize to arbitrary $β\ge 1$ the ones obtained for $β=2$ by the author, O'Connell and Warren between $h$-transformed Karlin-McGregor semigroups. Moreover they form the key step towards constructing a multilevel process in a Gelfand-Tsetlin pattern leaving certain Gibbs measures invariant. Finally as a by product we obtain a relation betw…
We derive an explicit expression for the kernel of the evolution group $\exp(-\mathrm{i} t H_0)$ of the discrete Laguerre operator $H_0$ (i.e. the Jacobi operator associated with the Laguerre polynomials) in terms of Jacobi polynomials. Based on this expression we show that the norm of the evolution group acting from $\ell^1$ to $\ell^\infty$ is given by $(1+t^2)^{-1/2}$.
The $β$-ensembles of random matrix theory with classical weights have many special properties. One is that the loop equations specifying the resolvent and corresponding multipoint correlators permit a derivation at general order of the correlator via Aomoto's method from the theory of the Selberg integral. We use Aomoto's method to derive the full hierarchy of loop equations for Laguerre and Jacobi $β$ ensembles, and use these to systematically construct the explicit form of the $1/N$ expansion at low orders. This allows us to give the explicit form of corrections to the global density, and al…
In a previous paper, we presented conjectures of the recurrence relations with constant coefficients for the multi-indexed orthogonal polynomials of Laguerre, Jacobi, Wilson and Askey-Wilson types. In this paper we present a proof for the Laguerre and Jacobi cases. Their bispectral properties are also discussed, which give a method to obtain the coefficients of the recurrence relations explicitly. This paper extends to the Laguerre and Jacobi cases the bispectral techniques recently introduced by Gómez-Ullate et al. to derive explicit expressions for the coefficients of the recurrence relation…
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 so…
A new recurrence relation for exceptional orthogonal polynomials is proposed, which holds for type 1, 2 and 3. As concrete examples, the recurrence relations are given for Xj-Hermite, Laguerre and Jacobi polynomials in j = 1,2 case.
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 X_m Laguerre or X_m Jacobi exceptional orthogonal polynomials. These potentials are isospectral to their usual counterparts and possess translationally shape invariance property.
Quasi-Hermitian Hamiltonians associated with exceptional orthogonal polynomials
利用点正则变换的方法，我们导出了一些严格可解的有理扩展量子哈密顿量，它们本质上是非厄米的，其束缚态波函数是Laguerre型或Jacobi型$X_1$例外正交多项式.借助坐标的虚移：$ e^{-αp} x e^{αp} = x+ i α$，这些汉密尔顿算符被证明是准埃尔米特算符和伪埃尔米特算符。事实证明，相应的能谱是完全真实的。
We construct energy-dependent potentials for which the Schroedinger equations admit solu- tions in terms of exceptional orthogonal polynomials. Our method of construction is based on certain point transformations, applied to the equations of exceptional Hermite, Jacobi and Laguerre polynomials. We present several examples of boundary-value problems with energy-dependent potentials that admit a discrete spectrum and the corresponding normalizable solutions in closed form.
The classical orthogonal polynomials are usually defined by the Rodrigues' formula. This paper refers to a fractional extension of the classical Hermite, Laguerre, Jacobi, Charlier, Meixner, Krawtchouk and Hahn polynomials. By means of the Caputo operator of fractional calculus, C-Hermite, C-Laguerre, C-Legndre and the C-Jacobi functions are defined and their representation in terms of the hypergeometric functions are provided. Also, by means of the Gray and Zhang fractional difference oparator, fractional Charlier, Meixner, Krawtchouk and Hahn functions are defined and their representation in…
