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This is the second part of a series of papers concerning Hilbert-Schmidt integral operators acting on the Hilbert spaces associated with elliptic Calogero-Moser type Hamiltonians. We present an explicit diagonalization of special Hilbert-Schmidt operators arising in the free A N−1 case, and a spectral structure analysis of the commuting family of Hilbert-Schmidt operators associated with the general A N−1 case. [ABSTRACT FROM AUTHOR]

In previous work we introduced and studied a function R(a+, a_, c; v, &vcaron;) that generalizes the hypergeometric function. In this paper we focus on a similarity-transformed function ε (a+, a_, γ; v, &vcaron;), with parameters γ ∈ C4 related to the couplings c ∈ C4 by a shift depending on a+, a_. We show that the ε-function is invariant under all maps γ → w(γ), with w in the Weyl group of type D4. Choosing a+, a_ positive and y, &vcaron; real, we obtain detailed information on the ¦Re v¦ → ∞ asymptotics of the ε-function. In particular, we explicitly determine the leading asymptotics in terms of plane waves and the c-function that implements the similarity R → &epsilon. [ABSTRACT FROM AUTHOR]

For generic parameters (a+, a_, c) ∈ (0, ∞)2 × we associate a Hubert space transform to the "relativistic" hypergeometric function R(a+, a_, c; v, v) studied in previous papers. Restricting the couplings c to a certain polytope, we show that the (renormalized) R-function kernel gives rise to an isometry from the even subspace of L²(R, &wcaron;(&vcaron;)d&vcaron; to the even subspace of L²(R, w(v)dv), where &wcaron;(&vcaron;) and w(v) are positive and even weight functions. We prove that the orthogonal complement of the range of this isometry is spanned by N ∈ N pairwise orthogonal functions. The latter are in essence Askey-Wilson polynomials, arising from the R-function by choosing &vcaron; = iκn, with κ0,…,κN-1 distinct negative numbers. The two commuting analytic difference operators acting on the variable v for which R is a joint eigenfunction, give rise to two commuting self-adjoint Hamiltonians on the even subspace of L²(R, w(v)dv). We explicitly determine the relation of the time-dependent scattering theory for these dynamics to their joint spectral transform. [ABSTRACT FROM AUTHOR]