| 1. |
- Atai, Farrokh, 1985-
(författare)
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A kernel function approach to exact solutions of Calogero-Moser-Sutherland type models
- 2016
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Doktorsavhandling (övrigt vetenskapligt/konstnärligt)abstract
- This Doctoral thesis gives an introduction to the concept of kernel functionsand their signicance in the theory of special functions. Of particularinterest is the use of kernel function methods for constructing exact solutionsof Schrodinger type equations, in one spatial dimension, with interactions governedby elliptic functions. The method is applicable to a large class of exactlysolvable systems of Calogero-Moser-Sutherland type, as well as integrable generalizationsthereof. It is known that the Schrodinger operators with ellipticpotentials have special limiting cases with exact eigenfunctions given by orthogonalpolynomials. These special cases are discussed in greater detail inorder to explain the kernel function methods with particular focus on the Jacobipolynomials and Jack polynomials.
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| 2. |
- Atai, Farrokh, 1985-, et al.
(författare)
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Deformed Calogero-Sutherland model and fractional Quantum Hall effect
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Annan publikation (övrigt vetenskapligt/konstnärligt)abstract
- The deformed Calogero-Sutherland (CS) model is a quantum integrable systemwith arbitrary numbers of two types of particles and reducing to the standard CSmodel in special cases. We show that a known collective field description of theCS model, which is based on conformal field theory (CFT), is actually a collectivefield description of the deformed CS model. This provides a natural application ofthe deformed CS model in Wen’s effective field theory of the fractional quantumHall effect (FQHE), with the two kinds of particles corresponding to electrons andquasi-hole excitations. In particular, we use known mathematical results aboutsuper Jack polynomials to obtain simple explicit formulas for the orthonormal CFTbasis proposed by van Elburg and Schoutens in the context of the FQHE.
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| 3. |
- Atai, Farrokh, et al.
(författare)
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Deformed Calogero-Sutherland model and fractional quantum Hall effect
- 2017
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Ingår i: Journal of Mathematical Physics. - : American Institute of Physics (AIP). - 0022-2488 .- 1089-7658. ; 58:1
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Tidskriftsartikel (refereegranskat)abstract
- The deformed Calogero-Sutherland (CS) model is a quantum integrable system with arbitrary numbers of two types of particles and reducing to the standard CS model in special cases. We show that a known collective field description of the CS model, which is based on conformal field theory (CFT), is actually a collective field description of the deformed CS model. This provides a natural application of the deformed CS model in Wen's effective field theory of the fractional quantum Hall effect (FQHE), with the two kinds of particles corresponding to electrons and quasi-hole excitations. In particular, we use known mathematical results about super-Jack polynomials to obtain simple explicit formulas for the orthonormal CFT basis proposed by van Elburg and Schoutens in the context of the FQHE.
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| 4. |
- Atai, Farrokh, et al.
(författare)
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Exact solutions by integrals of the non-stationary elliptic Calogero–Sutherland equation
- 2020
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Ingår i: Journal of Integrable Systems. - : Oxford University Press. - 2058-5985. ; 5:1
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Tidskriftsartikel (refereegranskat)abstract
- We use generalized kernel functions to construct explicit solutions by integrals of the non-stationary Schrödinger equation for the Hamiltonian of the elliptic Calogero–Sutherland model (also known as elliptic Knizhnik–Zamolodchikov–Bernard equation). Our solutions provide integral representations of elliptic generalizations of the Jack polynomials.
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| 5. |
- Atai, Farrokh, 1985-, et al.
(författare)
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Series solutions of the non-stationary Heun equation
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Annan publikation (övrigt vetenskapligt/konstnärligt)abstract
- We consider the non-stationary Heun equation, also known as quantum PainlevéVI, which has appeared in dierent works on quantum integrable models and conformaleld theory. We use a generalized kernel function identity to transform the problemto solve this equation into a dierential-dierence equation which, as we show, canbe solved by ecient recursive algorithms. We thus obtain series representations ofsolutions which provide elliptic generalizations of the Jacobi polynomials. These seriesreproduces, in a limiting case, a perturbative solution of the Heun equation due toTakemura, but our method is dierent in that we expand in non-conventional basisfunctions that allow us to obtain explicit formulas to all orders;
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| 6. |
- Atai, Farrokh, et al.
(författare)
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Series Solutions of the Non-Stationary Heun Equation
- 2018
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Ingår i: Symmetry, Integrability and Geometry. - : NATL ACAD SCI UKRAINE, INST MATH. - 1815-0659. ; 14
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Tidskriftsartikel (refereegranskat)abstract
- We consider the non-stationary Heun equation, also known as quantum Painleve VI, which has appeared in different works on quantum integrable models and conformal field theory. We use a generalized kernel function identity to transform the problem to solve this equation into a differential-difference equation which, as we show, can be solved by efficient recursive algorithms. We thus obtain series representations of solutions which provide elliptic generalizations of the Jacobi polynomials. These series reproduce, in a limiting case, a perturbative solution of the Heun equation due to Takemura, but our method is different in that we expand in non-conventional basis functions that allow us to obtain explicit formulas to all orders; in particular, for special parameter values, our series reduce to a single term.
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| 7. |
- Atai, Farrokh, et al.
(författare)
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Source Identities and Kernel Functions for Deformed (Quantum) Ruijsenaars Models
- 2014
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Ingår i: Letters in Mathematical Physics. - : Springer Science and Business Media LLC. - 0377-9017 .- 1573-0530. ; 104:7, s. 811-835
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Tidskriftsartikel (refereegranskat)abstract
- We consider the relativistic generalization of the quantum A (N-1) Calogero-Sutherland models due to Ruijsenaars, comprising the rational, hyperbolic, trigonometric and elliptic cases. For each of these cases, we find an exact common eigenfunction for a generalization of Ruijsenaars analytic difference operators that gives, as special cases, many different kernel functions; in particular, we find kernel functions for Chalykh-Feigin-Veselov-Sergeev-type deformations of such difference operators which generalize known kernel functions for the Ruijsenaars models. We also discuss possible applications of our results.
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| 8. |
- 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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| 9. |
- Atai, Farrokh, et al.
(författare)
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Variational orthogonalization
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Annan publikation (övrigt vetenskapligt/konstnärligt)abstract
- We introduce variational methods for finding approximate eigenfunctions and eigenvalues of quantum Hamiltonians by constructing a set of orthogonal wave functions which approximately solve the eigenvalue equation.
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| 10. |
- Berntson, Björn K., et al.
(författare)
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Elliptic soliton solutions of the spin non-chiral intermediate long-wave equation
- 2023
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Ingår i: Letters in Mathematical Physics. - : Springer Nature. - 0377-9017 .- 1573-0530. ; 113:3
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Tidskriftsartikel (refereegranskat)abstract
- We construct elliptic multi-soliton solutions of the spin non-chiral intermediate long-wave (sncILW) equation with periodic boundary conditions. These solutions are obtained by a spin-pole ansatz including a dynamical background term; we show that this ansatz solves the periodic sncILW equation provided the spins and poles satisfy the elliptic A-type spin Calogero-Moser (sCM) system with certain constraints on the initial conditions. The key to this result is a Backlund transformation for the elliptic sCM system which includes a non-trivial dynamical background term. We also present solutions of the sncILW equation on the real line and of the spin Benjamin-Ono equation which generalize previously obtained solutions by allowing for a non-trivial background term.
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