Product formulas for the relativistic and nonrelativistic conical functions

Hallnäs, Martin; Ruijsenaars, S. N. M.

doi:10.2969/aspm/07610195

Cited by 7 publications

(10 citation statements)

References 15 publications

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“…An explicit 1-variable example illustrating this hyperbolic conjecture can be found in Section 2 of [HaRu18], see in particular Eq. (92) on p. 213.…”

Section: Further Developments 41 the A 2 Casementioning

confidence: 97%

On Razamat's $A_2$ and $A_3$ kernel identities

Ruijsenaars

2020

Preprint

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In recent work on superconformal quantum field theories, Razamat arrived at elliptic kernel identities of a new and striking character: They relate solely to the root systems A 2 and A 3 and have no coupling type parameters [Ra18]. The pertinent 2and 3-variable Hamiltonians are analytic difference operators and the kernel functions are built from the elliptic gamma function. Razamat presented compelling evidence for the validity of these identities, and checked them to a certain order in a power series expansion. This paper is mainly concerned with analytical proofs of these identities. More specifically, we furnish a complete proof of the identities for the A 2 elliptic case and for the A 3 hyperbolic case, and consider several specializations. We also discuss the implications the kernel identities might have for a Hilbert space scenario involving common eigenvectors of the Hamiltonians and the integral operators associated with the kernel functions.

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“…An explicit 1-variable example illustrating this hyperbolic conjecture can be found in Section 2 of [HaRu18], see in particular Eq. (92) on p. 213.…”

Section: Further Developments 41 the A 2 Casementioning

confidence: 97%

On Razamat's $A_2$ and $A_3$ kernel identities

Ruijsenaars

2020

Preprint

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“…Then it follows from (1.29) and (3.4) that J 3 (b; x, y) is given by the left-hand side of (3.9). The crux is now that the F 2 -transforms of the functions f and g chosen above can be readily computed by using results from [HR15]. We proceed to embark on this.…”

Section: Invariance Properties and A Duality Relationmentioning

confidence: 99%

“…Keeping in mind (3.5) and (3.8), we infer from Eq. (3.24) in [HR15] the generalized eigenfunction expansion…”

Section: Invariance Properties and A Duality Relationmentioning

confidence: 99%

“…As a principal result of Subsection 3.1, we deduce a novel representation for J 3 , related to (1.29) by taking (b, x, y) → (2a − b, y, x). To generalize our approach in the N = 2 case, we rely on results from our recent joint paper [HR15] on product formulas for conical functions. Specifically, starting from the Plancherel relation for a generalized Fourier transform, we make use of the remarkable fact that J 2 (b; z, y) is an eigenfunction of the integral operator whose kernel is the product of the function…”

Section: Introductionmentioning

confidence: 99%

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Joint Eigenfunctions for the Relativistic Calogero–Moser Hamiltonians of Hyperbolic Type II. The Two- and Three-Variable Cases

Hallnäs

Ruijsenaars

2017

International Mathematics Research Notices

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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 → ∞ and y 1 − y 2 , y 2 − y 3 → ∞, resp., from which the conjectured factorized scattering can be read off.

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“…(1.8) (For the one-variable case, such S 3 -symmetric kernel identities have appeared before in several disparate contexts, see for example [Ko84,CMS99,Ru13,HaLa15,HaRu18].) The AΔO family is explicitly given by…”

Section: Introductionmentioning

confidence: 99%

On Razamat’s A ₂ and A ₃ kernel identities

Ruijsenaars¹

2020

J. Phys. A: Math. Theor.

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In recent work on superconformal quantum field theories, Razamat arrived at elliptic kernel identities of a new and striking character: they relate solely to the root systems A 2 and A 3 and have no coupling type parameters, Razamat S S (2019 Lett. Math. Phys. 109 1377-95). The pertinent two-and three-variable Hamiltonians are analytic difference operators and the kernel functions are built from the elliptic gamma function. Razamat presented compelling evidence for the validity of these identities, and checked them to a certain order in a power series expansion. This paper is mainly concerned with analytical proofs of these identities. More specifically, we furnish a complete proof of the identities for the A 2 elliptic case and for the A 3 hyperbolic case, and consider several specializations. We also discuss the implications the kernel identities might have for a Hilbert space scenario involving common eigenvectors of the Hamiltonians and the integral operators associated with the kernel functions.

show abstract

Product formulas for the relativistic and nonrelativistic conical functions

Cited by 7 publications

References 15 publications

On Razamat's $A_2$ and $A_3$ kernel identities

On Razamat's $A_2$ and $A_3$ kernel identities

Joint Eigenfunctions for the Relativistic Calogero–Moser Hamiltonians of Hyperbolic Type II. The Two- and Three-Variable Cases

On Razamat’s A ₂ and A ₃ kernel identities

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Product formulas for the relativistic and nonrelativistic conical functions

Cited by 7 publications

References 15 publications

On Razamat's $A_2$ and $A_3$ kernel identities

On Razamat's $A_2$ and $A_3$ kernel identities

Joint Eigenfunctions for the Relativistic Calogero–Moser Hamiltonians of Hyperbolic Type II. The Two- and Three-Variable Cases

On Razamat’s A 2 and A 3 kernel identities

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On Razamat’s A ₂ and A ₃ kernel identities