Multivariable Bessel polynomials related to the hyperbolic Sutherland model with external Morse potential

A multivariable generalization of the Bessel polynomials is introduced and studied. In particular, we deduce their series expansion in Jack polynomials, a limit transition from multivariable Jacobi polynomials, a sequence of algebraically independent eigenoperators, Pieri-type recurrence relations, and certain orthogonality properties.We also show that these multivariable Bessel polynomials provide a (finite) set of eigenfunctions of the hyperbolic Sutherland model with external Morse potential

An orthogonality relation for multivariable Bessel polynomials

In a recent paper we introduced a multivariable generalization of the Bessel polynomials, depending on one extra parameter, and related to the so-called hyperbolic Calogero–Moser–Sutherland model with external Morse potential. In this paper, we obtain a corresponding multivariable generalization of a well-known orthogonality relation for the (one-variable) Bessel polynomials due to Krall and Frink [H.L. Krall and O. Frink, A new class of orthogonal polynomials: the Bessel polynomials, Trans. Amer. Math. Soc. 65 (1949), pp. 100–115]

A recursive construction of joint eigenfunctions for the hyperbolic nonrelativistic Calogero-Moser Hamiltonians

We obtain symmetric joint eigenfunctions for the commuting partial differential operators associated to the hyperbolic Calogero-Moser N-particle system. The eigenfunctions are constructed via a recursion scheme, which leads to representations by multidimensional integrals whose integrands are elementary functions. We also tie in these eigenfunctions with the Heckman–Opdam hypergeometric function for the root system AN−1

A unified construction of generalized classical polynomials associated with operators of calogero-Sutherland Type

In this paper we consider a large class of many-variable polynomials which contains generalizations of the classical Hermite, Laguerre, Jacobi and Bessel polynomials as special cases, and which occur as the polynomial part in the eigenfunctions of Calogero–Sutherland type operators and their deformations recently found and studied by Chalykh, Feigin, Sergeev, and Veselov. We present a unified and explicit construction of all these polynomials

Explicit formulae for the eigenfunctions of the N-body Calogero model

We consider the quantum Calogero model, which describes N non-distinguishable quantum particles on the real line confined by a harmonic oscillator potential and interacting via two-body interactions proportional to the inverse square of the inter-particle distance. We elaborate a novel solution algorithm which allows us to obtain fully explicit formulae for its eigenfunctions, arbitrary coupling parameter and particle number. We also show that our method applies, with minor changes, to all Calogero models associated with classical root systems

On the spectra of real and complex Lame operators

On the spectra of real and complex Lame operator

Complex exceptional orthogonal polynomials and quasi-invariance

Consider the Wronskians of the classical Hermite polynomials Hλ₁(x):= Wr(Hl(x);Hk1 (x)…;Hkn(x)); l ϵ Z≥0 \{k1; : : : ; kn}; where ki = λ₁ + n - i; i = 1;…, n and λ = (λ₁;…; λn) is a partition. Gómez-Ullate et al. showed that for a special class of partitions the corresponding polynomials are orthogonal and dense among all polynomials with respect to a certain inner product, but in contrast to the usual case have some degrees missing (so called exceptional orthogonal polynomials). We generalise their results to all partitions by considering complex contours of integration and non-positive Hermitian products. The corresponding polynomials are orthogonal and dense in a finite-codimensional subspace of C[x] satisfying certain quasi-invariance conditions. A Laurent version of exceptional orthogonal polynomials, related to monodromy-free trigonometric Schrödinger operators, is also presented

Complex exceptional orthogonal polynomials and quasi-invariance

Consider the Wronskians of the classical Hermite polynomials Hλ₁(x):= Wr(Hl(x);Hk1 (x)…;Hkn(x)); l ϵ Z≥0 \{k1; : : : ; kn}; where ki = λ₁ + n - i; i = 1;…, n and λ = (λ₁;…; λn) is a partition. Gómez-Ullate et al. showed that for a special class of partitions the corresponding polynomials are orthogonal and dense among all polynomials with respect to a certain inner product, but in contrast to the usual case have some degrees missing (so called exceptional orthogonal polynomials). We generalise their results to all partitions by considering complex contours of integration and non-positive Hermitian products. The corresponding polynomials are orthogonal and dense in a finite-codimensional subspace of C[x] satisfying certain quasi-invariance conditions. A Laurent version of exceptional orthogonal polynomials, related to monodromy-free trigonometric Schrödinger operators, is also presented

Exact solutions of two complementary one-dimensional quantum many-body systems on the half-line

We consider two particular one-dimensional quantum many-body systems with local interactions related to the root system CN. Both models describe identical particles moving on the half-line with nontrivial boundary conditions at the origin, but in the first model the particles interact with the delta interaction while in the second via a particular momentum dependent interaction commonly known as delta-prime interaction. We show that the Bethe ansatz solution of the delta-interaction model is consistent even for the general case where the particles are distinguishable, whereas for the delta-prime interaction it only is consistent and nontrivial in the fermion case. We also establish a duality between the bosonic delta- and the fermionic delta-prime model, and we elaborate on the physical interpretations of these models

Kernel functions Backlund transformations for relativistic Calogero-Moser Toda systems

We obtain kernel functions associated with the quantum relativistic Toda systems, both for the periodic version and for the nonperiodic version with its dual. This involves taking limits of previously known results concerning kernel functions for the elliptic and hyperbolic relativistic Calogero-Moser systems. We show that the special kernel functions at issue admit a limit that yields generating functions of Bäcklund transformations for the classical relativistic Calogero-Moser and Toda systems. We also obtain the nonrelativistic counterparts of our results, which tie in with previous results in the literature