Finite lattice Bethe ansatz systems and the Heun equation

Dorey, Patrick; Suzuki, Junji; Tateo, Roberto

doi:10.1088/0305-4470/37/6/006

Cited by 23 publications

(29 citation statements)

References 35 publications

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“…The intimate connection between Heun's equation and GAL potentials is well known [9]. In recent years, this equation has also proved very useful in the context of a number of physical problems, like quasi-exactly solvable systems [23], sphaleron stability [24], Calogero-Sutherland models [25], higher dimensional correlated systems [26], Kerr-de Sitter black holes [27], and finite lattice Bethe ansatz systems [28].…”

mentioning

confidence: 99%

Complex periodic potentials with a finite number of band gaps

Khare

Sukhatme

2006

Journal of Mathematical Physics

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We obtain several new results for the complex generalized associated Lamé potential, sn(y, m) is a Jacobi elliptic function with modulus parameter m, and there are four real parameters a, b, f, g. First, we derive two new duality relations which, when coupled with a previously obtained duality relation, permit us to relate the band edge eigenstates of the 24 potentials obtained by permutations of the four parameters a, b, f, g. Second, we pose and answer the question: how many independent potentials are there with a finite number "a" of band gaps when a, b, f, g are integers?For these potentials, we clarify the nature of the band edge eigenfunctions. We also obtain several analytic results when at least one of the four parameters is a half-integer. As a by-product, we also obtain new solutions of Heun's differential equation.1

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confidence: 99%

Complex periodic potentials with a finite number of band gaps

Khare

Sukhatme

2006

Journal of Mathematical Physics

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“…In this appendix, we take a formal but simple approach to represent the solution ϕ(x): we try to represent it with a familiar object, a limit of the generalized hypergeometric series 3 F 2 . The following argument generalizes the one given for the Airy function in its relation to finite size lattice models [38].…”

Section: Appendix B Solutions To the 3 Rd Order Odementioning

confidence: 73%

“…The parameter N corresponds to the lattice size for the SU(2) case [38]. The desired solution, ϕ, however should satisfy the asymptotic behavior (2).…”

Section: Appendix B Solutions To the 3 Rd Order Odementioning

confidence: 99%

Elementary functions in thermodynamic Bethe ansatz

Suzuki¹

2015

J. Phys. A: Math. Theor.

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Some years ago, Fendley found an explicit solution to the thermodynamic Bethe ansatz (TBA) equation for a N = 2 supersymmetric theory in 2D with a specific F-term. Motivated by this, we seek for explicit solutions for other super-potential cases utilizing the idea from the ODE/IM correspondence. We find that the TBA equations, corresponding to a wider class of super-potentials, admit solutions in terms of elementary functions such as modified Bessel functions and confluent hyper-geometric series. * Based on talks given at "Infinite Analysis 2014" (Tokyo, Feb. 2014) and at "Integrable lattice models and quantum field theories" (Bad-Honnef, June 2014) †

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“…o In a relatively recent work P. Dorey, [55] showed that equations in finite lattice systems also reduce to Heun equations. [56].…”

Section: Some Examples Of the Heun Equation In Physical Applicationsmentioning

confidence: 99%

Heun Functions and Some of Their Applications in Physics

Hortaçsu

2018

Advances in High Energy Physics

102

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Most of the theoretical physics known today is described by using a small number of differential equations. For linear systems, different forms of the hypergeometric or the confluent hypergeometric equations often suffice to describe the system studied. These equations have power series solutions with simple relations between consecutive coefficients and/ or can be represented in terms of simple integral transforms. If the problem is nonlinear, one often uses one form of the Painlevé equations. There are important examples, however, where one has to use higher order equations. Heun equation is one of these examples, which recently is often encountered in problems in general relativity and astrophysics. Its special and confluent forms take names as Mathieu, Lamé and Coulomb spheroidal equations. For these equations whenever a power series solution is written, instead of a two way recursion relation between the coefficients in the series, we find one between three or four different ones. An integral transform solution using simpler functions also is not obtainable.The use of this equation in physics and mathematical literature exploded in the later years, more than doubling the number of papers with these solutions in the last decade, compared to time period since this equation was introduced in 1889 up to 2008. We use SCI data to conclude this statement, which is not precise, but in the correct ballpark. Here this equation will be introduced and examples for its use, especially in general relativity literature will be given. * This paper is a revised and many times updated version of the conference talk by the same author, published in

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Finite lattice Bethe ansatz systems and the Heun equation

Cited by 23 publications

References 35 publications

Complex periodic potentials with a finite number of band gaps

Complex periodic potentials with a finite number of band gaps

Elementary functions in thermodynamic Bethe ansatz

Heun Functions and Some of Their Applications in Physics

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