| 1. |
- Atai, F., et al.
(författare)
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Super-Macdonald Polynomials: Orthogonality and Hilbert Space Interpretation
- 2021
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Ingår i: Communications in Mathematical Physics. - : Springer Science and Business Media LLC. - 0010-3616 .- 1432-0916. ; 388:1, s. 435-468
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Tidskriftsartikel (refereegranskat)abstract
- The super-Macdonald polynomials, introduced by Sergeev andVeselov (Commun Math Phys 288: 653-675, 2009), 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 in Sergeev and Veselov (CommunMath Phys 245: 249-278, 2004). 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 MacdonaldRuijsenaars operators. This allows for a quantum mechanical interpretation of the models defined by the deformedMacdonald-Ruijsenaars operators. Motivated by recent results in the nonrelativistic (q -> 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.
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| 2. |
- Farrokh, Atai, et al.
(författare)
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Orthogonality of super‐Jack polynomials and a Hilbert space interpretation of deformed Calogero–Moser–Sutherland operators
- 2019
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Ingår i: Bulletin of the London Mathematical Society. - : Wiley. - 0024-6093 .- 1469-2120. ; 51:2, s. 353-370
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Tidskriftsartikel (refereegranskat)abstract
- We prove orthogonality and compute explicitly the (quadratic) norms for super-Jack polynomials SP lambda((z1, horizontal ellipsis ,zn),(w1, horizontal ellipsis ,wm);theta) with respect to a natural positive semi-definite, but degenerate, Hermitian product ⟨center dot,center dot⟩n,m,theta '. In case m=0 (or n=0), our product reduces to Macdonald's well-known inner product ⟨center dot,center dot⟩n,theta ', and we recover his corresponding orthogonality results for the Jack polynomials P lambda((z1, horizontal ellipsis ,zn);theta). From our main results, we readily infer that the kernel of ⟨center dot,center dot⟩n,m,theta ' is spanned by the super-Jack polynomials indexed by a partition lambda not containing the mxn rectangle (mn). As an application, we provide a Hilbert space interpretation of the deformed trigonometric Calogero-Moser-Sutherland operators of type A(n-1,m-1).
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| 3. |
- Feigin, Misha V., et al.
(författare)
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Quasi-invariant Hermite Polynomials and Lassalle-Nekrasov Correspondence
- 2021
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Ingår i: Communications in Mathematical Physics. - : Springer Science and Business Media LLC. - 1432-0916 .- 0010-3616. ; 386:1, s. 107-141
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Tidskriftsartikel (refereegranskat)abstract
- Lassalle and Nekrasov discovered in the 1990s a surprising correspondence between the rational Calogero-Moser system with a harmonic term and its trigonometric version. We present a conceptual explanation of this correspondence using the rational Cherednik algebra and establish its quasi-invariant extension. More specifically, we consider configurations A of real hyperplanes with multiplicities admitting the rational Baker-Akhiezer function and use this to introduce a new class of non-symmetric polynomials, which we call A-Hermite polynomials. These polynomials form a linear basis in the space of A-quasi-invariants, which is an eigenbasis for the corresponding generalised rational Calogero-Moser operator with harmonic term. In the case of the Coxeter configuration of type AN this leads to a quasi-invariant version of the Lassalle-Nekrasov correspondence and its higher order analogues.
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| 4. |
- Görbe, Tamas, et al.
(författare)
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Quantization and explicit diagonalization of new compactified trigonometric Ruijsenaars–Schneider systems
- 2018
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Ingår i: Journal of Integrable Systems. - : Oxford University Press (OUP). - 2058-5985. ; 3:1
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Tidskriftsartikel (refereegranskat)abstract
- Recently, Fehér and Kluck discovered, at the level of classical mechanics, new compactified trigonometric Ruijsenaars–Schneider n -particle systems, with phase space symplectomorphic to the (n−1) -dimensional complex projective space. In this article, we quantize the so-called type (i) instances of these systems and explicitly solve the joint eigenvalue problem for the corresponding quantum Hamiltonians by generalising previous results of van Diejen and Vinet. Specifically, the quantum Hamiltonians are realized as discrete difference operators acting in a finite-dimensional Hilbert space of complex-valued functions supported on a uniform lattice over the classical configuration space, and their joint eigenfunctions are constructed in terms of discretized An−1 Macdonald polynomials with unitary parameters.
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| 5. |
- Haese-Hill, W. A., et al.
(författare)
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On the Spectra of Real and Complex Lamé Operators
- 2017
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Ingår i: Symmetry Integrability and Geometry-Methods and Applications. - 1815-0659. ; 13
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Tidskriftsartikel (refereegranskat)abstract
- We study Lame operators of the form with m is an element of N and omega a half- period of P(z). For rectangular period lattices, we can choose omega and z(0) such that the potential is real, periodic and regular. It is known after Ince that the spectrum of the corresponding Lame operator has a band structure with not more than m gaps. In the first part of the paper, we prove that the opened gaps are precisely the first m ones. In the second part, we study the Lame spectrum for a generic period lattice when the potential is complex- valued. We concentrate on the m = 1 case, when the spectrum consists of two regular analytic arcs, one of which extends to infinity, and briefly discuss the m = 2 case, paying particular attention to the rhombic lattices.
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| 6. |
- Hallnäs, Martin, 1979
(författare)
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Baxter Q-operator for the hyperbolic Calogero-Moser system
- 2024
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Ingår i: Journal of Physics A: Mathematical and Theoretical. - 1751-8121 .- 1751-8113. ; 57:22
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Tidskriftsartikel (refereegranskat)abstract
- We introduce a Q-operator 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.
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| 7. |
- Hallnäs, Martin, 1979, et al.
(författare)
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From Kajihara’s transformation formula to deformed Macdonald–Ruijsenaars and Noumi–Sano operators
- 2022
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Ingår i: Selecta Mathematica, New Series. - : Springer Science and Business Media LLC. - 1420-9020 .- 1022-1824. ; 28:2
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Tidskriftsartikel (refereegranskat)abstract
- Kajihara obtained in 2004 a remarkable transformation formula connecting multiple basic hypergeometric series associated with A-type root systems of different ranks. By 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.
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| 8. |
- Hallnäs, Martin, 1979, et al.
(författare)
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Higher Order Deformed Elliptic Ruijsenaars Operators
- 2022
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Ingår i: Communications in Mathematical Physics. - : Springer Science and Business Media LLC. - 0010-3616 .- 1432-0916. ; 392, s. 659-689
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Tidskriftsartikel (refereegranskat)abstract
- 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.
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| 9. |
- Hallnäs, Martin, 1979, et al.
(författare)
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Joint Eigenfunctions for the Relativistic Calogero–Moser Hamiltonians of Hyperbolic Type II. The Two- and Three-Variable Cases
- 2018
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Ingår i: International mathematics research notices. - : Oxford University Press (OUP). - 1073-7928 .- 1687-0247. ; :14, s. 4404-4449
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Tidskriftsartikel (refereegranskat)abstract
- 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 $\rE_2(b;x,y)$ and $\rE_3(b;x,y)$. 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.
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| 10. |
- Hallnäs, Martin, 1979, et al.
(författare)
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Joint Eigenfunctions for the Relativistic Calogero–Moser Hamiltonians of Hyperbolic Type. III. Factorized Asymptotics
- 2021
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Ingår i: International Mathematics Research Notices. - : Oxford University Press (OUP). - 1073-7928 .- 1687-0247. ; 2021:6, s. 4679-4708
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Tidskriftsartikel (refereegranskat)abstract
- 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 $N>3$.
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| 11. |
- Hallnäs, Martin, 1979
(författare)
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New Orthogonality Relations for Super-Jack Polynomials and an Associated Lassalle-Nekrasov Correspondence
- 2023
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Ingår i: Constructive Approximation. - : Springer Science and Business Media LLC. - 0176-4276 .- 1432-0940.
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Tidskriftsartikel (refereegranskat)abstract
- The super-Jack polynomials, introduced by Kerov, Okounkov and Olshanski, are polynomials inn+m variables, which reduce to the Jack polynomials when n = 0 or m = 0 and provide joint eigenfunctions of the quantum integrals of the deformed trigonometric Calogero-Moser-Sutherland system. We prove that the super-Jack polynomials are orthogonal with respect to a bilinear form of the form (p, q) (SIC)(L(p)q)(0), with L-p quantum integrals of the deformed rational Calogero-Moser-Sutherland system. In addition, we provide a new proof of the Lassalle-Nekrasov correspondence between deformed trigonometric and rational harmonic Calogero-Moser-Sutherland systems and infer orthogonality of super-Hermite polynomials, which provide joint eigenfunctions of the latter system.
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| 12. |
- Hallnäs, Martin, 1979, et al.
(författare)
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Product formulas for the relativistic and nonrelativistic conical functions
- 2018
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Ingår i: Representation theory, special functions and Painlevé equations. RIMS 2015, Kyoto, Japan, March 3-6, 2015, s. 195-245. - : Mathematical Society of Japan / World Scientific Publishing. ; 76, s. 195-245
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Konferensbidrag (refereegranskat)abstract
- 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.
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| 13. |
- Hallnäs, Martin, 1979-
(författare)
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Quantum many-body systems exactly solved by special functions
- 2007
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Doktorsavhandling (övrigt vetenskapligt/konstnärligt)abstract
- This thesis concerns two types of quantum many-body systems in one dimension exactly solved by special functions: firstly, systems with interactions localised at points and solved by the (coordinate) Bethe ansatz; secondly, systems of Calogero-Sutherland type, as well as certain recently introduced deformations thereof, with eigenfunctions given by natural many-variable generalisations of classical (orthogonal) polynomials. The thesis is divided into two parts. The first provides background and a few complementary results, while the second presents the main results of this thesis in five appended scientific papers. In the first paper we consider two complementary quantum many-body systems with local interactions related to the root systems CN, one with delta-interactions, and the other with certain momentum dependent interactions commonly known as delta-prime interactions. We prove, by construction, that the former is exactly solvable by the Bethe ansatz in the general case of distinguishable particles, and that the latter is similarly solvable only in the case of bosons or fermions. We also establish a simple strong/weak coupling duality between the two models and elaborate on their physical interpretations. In the second paper we consider a well-known four-parameter family of local interactions in one dimension. In particular, we determine all such interactions leading to a quantum many-body system of distinguishable particles exactly solvable by the Bethe ansatz. We find that there are two families of such systems: the first is described by a one-parameter deformation of the delta-interaction model, while the second features a particular one-parameter combination of the delta and the delta-prime interactions. In papers 3-5 we construct and study particular series representations for the eigenfunctions of a family of Calogero-Sutherland models naturally associated with the classical (orthogonal) polynomials. In our construction, the eigenfunctions are given by linear combinations of certain symmetric polynomials generalising the so-called Schur polynomials, with explicit and rather simple coefficients. In paper 5 we also generalise certain of these results to the so-called deformed Calogero-Sutherland operators.
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