arXiv subjects

Martin Hallnäs

Publications and source records attributed to Martin Hallnäs.

At least 19 recordsLinked to original sources

arXiv2026

Baker-Akhiezer specialisation of joint eigenfunctions for hyperbolic relativistic Calogero-Moser Hamiltonians

In earlier joint work with Ruijsenaars, we constructed and studied symmetric joint eigenfunctions $J_N$ for quantum Hamiltonians of the hyperbolic relativistic $N$-particle Calogero--Moser system. For generic coupling values, they are non-elementary functions that in the $N=2$ case essentially amount to a `relativistic' generalisation of the conical function specialisation of the Gauss hypergeometric function ${}_2F_1$. In this paper, we consider a discrete set of coupling values for which the solution to the joint eigenvalue problem is known to be given by functions $ψ_N$ of Baker--Akhiezer type, which are elementary, but highly nontrivial, functions. Specifically, we show that $J_N$ essentially amounts to the antisymmetrisation of $ψ_N$ and, as a byproduct, we obtain a recursive construction of $ψ_N$ in terms of an iterated residue formula.

math.QA↗

arXiv2024

Elliptic Integrable Systems and Special Functions

We discuss elliptic quantum Calogero-Moser-Sutherland models, including their relativistic generalizations due to Ruijsenaars and van Diejen, and the relations of these models to classes of special functions developed and explored in recent and on-going research efforts. We give an introduction to these models and corresponding special functions aimed at non-experts. We also describe a few open problems in the field.

math-ph↗

arXiv2024

Calogero-Moser-Sutherland systems

We discuss integrable many-body systems in one dimension of Calogero-Moser-Sutherland type, both classical and quantum as well as nonrelativistic and relativistic. In particular, we consider fundamental properties such as integrability, the existence of explicit solutions as well as action-angle and bispectral dualities that relate different such systems. We also briefly discuss the early history of the subject and indicate connections with other integrable systems.

math-ph↗

arXiv2023

Baxter $Q$-operator for the hyperbolic Calogero--Moser system

We introduce a $Q$-operator $\mathcal{Q}_z$ for the hyperbolic Calogero--Moser system as a one-parameter family of explicit integral operators. We establish the standard properties of a $Q$-operator, i.e.~invariance of Hamiltonians, commutativity for different parameter values and that its eigenvalues satisfy an explicitly given first order ordinary difference equation in the parameter $z$.

math-ph↗

arXiv2023

New orthogonality relations for super-Jack polynomials and an associated Lassalle--Nekrasov correspondence

The super-Jack polynomials, introduced by Kerov, Okounkov and Olshanski, are polynomials in $n+m$ variables, which reduce to the Jack polynomials when $n=0$ or $m=0$ and provide joint eigenfunctions of the quantum integrals of the trigonometric deformed Calogero-Moser-Sutherland system. We prove that the super-Jack polynomials are orthogonal with respect to a bilinear form of the form $(p,q)\mapsto (L_pq)(0)$, with $L_p$ quantum integrals of the rational deformed Calogero-Moser-Sutherland system. In addition, we provide a new proof of the Lassalle-Nekrasov correspondence between trigonometric and rational harmonic deformed Calogero-Moser-Sutherland systems and infer orthogonality of super-Hermite polynomials, which provide joint eigenfunctions of the latter system.

math.QA↗

arXiv2021

Higher order deformed elliptic Ruijsenaars operators

We present four infinite families of mutually commuting difference operators which include the deformed elliptic Ruijsenaars operators. The trigonometric limit of this kind of operators was previously introduced by Feigin and Silantyev. They provide a quantum mechanical description of two kinds of relativistic quantum mechanical particles which can be identified with particles and anti-particles in an underlying quantum field theory. We give direct proofs of the commutativity of our operators and of some other fundamental properties such as kernel function identities. In particular, we give a rigorous proof of the quantum integrability of the deformed Ruijsenaars model.

math-ph↗

arXiv2021

From Kajihara's transformation formula to deformed Macdonald-Ruijsenaars and Noumi-Sano operators

Kajihara obtained in 2004 a remarkable transformation formula connecting multiple basic hypergeometric series associated with $A$-type root systems of different ranks. By multiple principle specialisations of his formula, we deduce kernel identities for deformed Macdonald-Ruijsenaars (MR) and Noumi-Sano (NS) operators. The deformed MR operators were introduced by Sergeev and Veselov in the first order case and by Feigin and Silantyev in the higher order cases. As applications of our kernel identities, we prove that all of these operators pairwise commute and are simultaneously diagonalised by the super-Macdonald polynomials. We also provide an explicit description of the algebra generated by the deformed MR and/or NS operators by a Harish-Chandra type isomorphism and show that the deformed MR (NS) operators can be viewed as restrictions of inverse limits of ordinary MR (NS) operators.

math.QA↗

arXiv2021

Super-Macdonald polynomials: Orthogonality and Hilbert space interpretation

The super-Macdonald polynomials, introduced by Sergeev and Veselov, generalise the Macdonald polynomials to (arbitrary numbers of) two kinds of variables, and they are eigenfunctions of the deformed Macdonald-Ruijsenaars operators introduced by the same authors. We introduce a Hermitian form on the algebra spanned by the super-Macdonald polynomials, prove their orthogonality, compute their (quadratic) norms explicitly, and establish a corresponding Hilbert space interpretation of the super-Macdonald polynomials and deformed Macdonald-Ruijsenaars operators. This allows for a quantum mechanical interpretation of the models defined by the deformed Macdonald-Ruijsenaars operators. Motivated by recent results in the nonrelativistic ($q\to 1$) case, we propose that these models describe the particles and anti-particles of an underlying relativistic quantum field theory, thus providing a natural generalisation of the trigonometric Ruijsenaars model.

math.QA↗

arXiv2019

Joint eigenfunctions for the relativistic Calogero-Moser Hamiltonians of hyperbolic type. III. Factorized asymptotics

In the two preceding parts of this series of papers, we introduced and studied a recursion scheme for constructing joint eigenfunctions $J_N(a_+, a_-,b;x,y)$ of the Hamiltonians arising in the integrable $N$-particle systems of hyperbolic relativistic Calogero-Moser type. We focused on the first steps of the scheme in Part I, and on the cases $N=2$ and $N=3$ in Part II. In this paper, we determine the dominant asymptotics of a similarity transformed function $\rE_N(b;x,y)$ for $y_j-y_{j+1}\to\infty$, $j=1,\ldots, N-1$, and thereby confirm the long standing conjecture that the particles in the hyperbolic relativistic Calogero-Moser system exhibit soliton scattering. This result generalizes a main result in Part II to all particle numbers $N>3$.

math-ph↗

arXiv2018

Orthogonality of super-Jack polynomials and a Hilbert space interpretation of deformed Calogero-Moser-Sutherland operators

We prove orthogonality and compute explicitly the (quadratic) norms for super-Jack polynomials $SP_λ((z_1,\ldots,z_n),(w_1,\ldots,w_m);θ)$ with respect to a natural positive semi-definite, but degenerate, Hermitian product $\langle\cdot,\cdot\rangle_{n,m}^\prime$. In case $m=0$ (or $n=0$), our product reduces to Macdonald's well-known inner product $\langle\cdot,\cdot\rangle_n^\prime$, and we recover his corresponding orthogonality results for the Jack polynomials $P_λ((z_1,\ldots,z_n);θ)$. From our main results, we readily infer that the kernel of $\langle\cdot,\cdot\rangle_{n,m}^\prime$ is spanned by the super-Jack polynomials indexed by a partition $λ$ not containing the $m\times n$ rectangle $(m^n)$. As an application, we provide a Hilbert space interpretation of the deformed trigonometric Calogero-Moser-Sutherland operators of type $A(n-1,m-1)$.

math.QA↗

arXiv2017

Joint eigenfunctions for the relativistic Calogero-Moser Hamiltonians of hyperbolic type. II. The two- and three-variable cases

In a previous paper we introduced and developed a recursive construction of joint eigenfunctions $J_N(a_+,a_-,b;x,y)$ for the Hamiltonians of the hyperbolic relativistic Calogero-Moser system with arbitrary particle number $N$. In this paper we focus on the cases $N=2$ and $N=3$, and establish a number of conjectured features of the corresponding joint eigenfunctions. More specifically, choosing $a_+,a_-$ positive, we prove that $J_2(b;x,y)$ and $J_3(b;x,y)$ extend to globally meromorphic functions that satisfy various invariance properties as well as a duality relation. We also obtain detailed information on the asymptotic behavior of similarity transformed functions E$_2(b;x,y)$ and E$_3(b;x,y)$. In particular, we determine the dominant asymptotics for $y_1-y_2\to\infty$ and $y_1-y_2,y_2-y_3\to\infty$, resp., from which the conjectured factorized scattering can be read off.

math-ph↗

arXiv2016

A product formula for the eigenfunctions of a quartic oscillator

We consider the Schrödinger operator on the real line with an even quartic potential. Our main result is a product formula of the type $ψ_k(x)ψ_k(y) = \int_{\mathbb{R}} ψ_k(z)\mathcal{K}(x,y,z)dz$ for its eigenfunctions $ψ_k$. The kernel function $\mathcal{K}$ is given explicitly in terms of the Airy function $\mathrm{Ai}(x)$, and is positive for appropriate parameter values. As an application, we obtain a particular asymptotic expansion of the eigenfunctions $ψ_k$.

math-ph↗

arXiv2016

Product formulas for the relativistic and nonrelativistic conical functions

The conical function and its relativistic generalization can be viewed as eigenfunctions of the reduced 2-particle Hamiltonians of the hyperbolic Calogero-Moser system and its relativistic generalization. We prove new product formulas for these functions. As a consequence, we arrive at explicit diagonalizations of integral operators that commute with the 2-particle Hamiltonians and reduced versions thereof. The kernels of the integral operators are expressed as integrals over products of the eigenfunctions and explicit weight functions. The nonrelativistic limits are controlled by invoking novel uniform limit estimates for the hyperbolic gamma function.

math.CA↗

arXiv2014

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

We obtain symmetric joint eigenfunctions for the commuting PDOs 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 A_{N-1}.

nlin.SI↗

arXiv2012

Kernel functions and Bäcklund transformations for relativistic Calogero-Moser and 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.

nlin.SI↗

arXiv2012

Hermite and Laguerre Symmetric Functions Associated with Operators of Calogero-Moser-Sutherland Type

We introduce and study natural generalisations of the Hermite and Laguerre polynomials in the ring of symmetric functions as eigenfunctions of infinite-dimensional analogues of partial differential operators of Calogero-Moser-Sutherland (CMS) type. In particular, we obtain generating functions, duality relations, limit transitions from Jacobi symmetric functions, and Pieri formulae, as well as the integrability of the corresponding operators. We also determine all ideals in the ring of symmetric functions that are spanned by either Hermite or Laguerre symmetric functions, and by restriction of the corresponding infinite-dimensional CMS operators onto quotient rings given by such ideals we obtain so-called deformed CMS operators. As a consequence of this restriction procedure, we deduce, in particular, infinite sets of polynomial eigenfunctions, which we shall refer to as super Hermite and super Laguerre polynomials, as well as the integrability, of these deformed CMS operators. We also introduce and study series of a generalised hypergeometric type, in the context of both symmetric functions and 'super' polynomials.

math.QA↗

arXiv2012

Joint eigenfunctions for the relativistic Calogero-Moser Hamiltonians of hyperbolic type. I. First steps

We present and develop a recursion scheme to construct joint eigenfunctions for the commuting analytic difference operators associated with the integrable N-particle systems of hyperbolic relativistic Calogero-Moser type. The scheme is based on kernel identities we obtained in previous work. In this first paper of a series we present the formal features of the scheme, show explicitly its arbitrary-N viability for the `free' cases, and supply the analytic tools to prove the joint eigenfunction properties in suitable holomorphy domains.

nlin.SI↗

arXiv2009

A unified construction of generalised classical polynomials associated with operators of Calogero-Sutherland type

In this paper we consider a large class of many-variable polynomials which contains generalisations 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.

math-ph↗