Source Identity and Kernel Functions for Elliptic Calogero–Sutherland Type Systems

Abstract

Kernel functions related to quantum many-body systems of Calogero-Sutherland type are discussed, in particular for the elliptic case. The main result is an elliptic generalization of an identity due to Sen that is a source for many such kernel functions. Applications are given, including simple exact eigenfunctions and corresponding eigenvalues of Chalykh-Feigin-Veselov-Sergeev-type deformations of the elliptic Calogero-Sutherland model for special parameter values. MSC-class: 81Q05, 16R60

Source identity and kernel functions for Inozemtsev-type systems

“…Source identities for all quantum Calogero-Moser-Sutherland models where obtained and used to derive kernel functions in [6]. A source identity allowing to derive kernel functions for the elliptic generalizations of the Sutherland model and their deformations was presented in [29]. The present paper generalizes, to the elliptic case, results previously obtained in [6].…”

Deformed Calogero-Sutherland model and fractional quantum Hall effect

“…It is obvious that Ψ N,M 0 (x, y; g) in (2.3) is invariant if all variables x j and y k are shifted by the same constant. Thus D N,M Ψ N,M 0 in (2.6b) can be obtained from Proposition 2.1 in[42] as the special case q = 0, N = N + M, m J = 1 for 1 ≤ J ≤ N, and m J = −1/λ for N + 1 ≤ J ≤ N + M (note that g here corresponds to λ in[42]).…”

Exact solutions by integrals of the non-stationary elliptic Calogero–Sutherland equation

“…We stress that (32) is formal since we neither specified the integration domain nor the parameters Q, Q ′ , and these details are important to get well-defined solutions. However, if we ignore such details for now, we can proceed with formal computations using (27)- (29) and dropping boundary terms obtained by partial integrations used to transfer the action of the differential operator (details on this computation are given in the proof of Lemma 4.2 below). We thus find that ψ N (x) = (K QQ ′ N M ψ M )(x) satisfies (30) with…”

“…We are left to prove that, if ψ M (y) satisfies (29), then ψ N (x) in (32) satisfies (30). To simplify notation, we write in the following short for [−π−iǫ,π−iǫ] M , and we use the short hand notations |x| = j x j and similarly for |y|.…”

“…This operator is known to define a quantum integrable system (this is proved in [28] for the caseñ = 1, but the result is expected to be true for arbitraryñ). Moreover, generalized kernel functions K N,Ñ ,M,M (x,x, y,ỹ) for this non-stationary deformed eCS equation are known [29]: they obey relations as in (28) but for deformed eCS operators H N,Ñ (x,x) and H M,M (y,ỹ), and κ = (N − M)g is replaced by (N − M)g −Ñ +M . We expect that, using these generalized kernel functions, the construction in the present paper can be generalized to find solutions of the deformed non-stationary eCS equation.…”