66 research outputs found
Joint eigenfunctions for the relativistic Calogero-Moser Hamiltonians of hyperbolic type. III. Factorized asymptotics
In the two preceding parts of this series of papers, we introduced and
studied a recursion scheme for constructing joint eigenfunctions of the Hamiltonians arising in the integrable -particle systems
of hyperbolic relativistic Calogero-Moser type. We focused on the first steps
of the scheme in Part I, and on the cases and in Part II. In this
paper, we determine the dominant asymptotics of a similarity transformed
function \rE_N(b;x,y) for , , and
thereby confirm the long standing conjecture that the particles in the
hyperbolic relativistic Calogero-Moser system exhibit soliton scattering. This
result generalizes a main result in Part II to all particle numbers .Comment: 21 page
A Relativistic Conical Function and its Whittaker Limits
In previous work we introduced and studied a function that is a generalization of the hypergeometric function
and the Askey-Wilson polynomials. When the coupling vector is specialized to , , we obtain
a function that generalizes the
conical function specialization of and the -Gegenbauer
polynomials. The function is the joint eigenfunction of four
analytic difference operators associated with the relativistic Calogero-Moser
system of type, whereas the function corresponds to , and is
the joint eigenfunction of four hyperbolic Askey-Wilson type difference
operators. We show that the -function admits five novel integral
representations that involve only four hyperbolic gamma functions and plane
waves. Taking their nonrelativistic limit, we arrive at four representations of
the conical function. We also show that a limit procedure leads to two
commuting relativistic Toda Hamiltonians and two commuting dual Toda
Hamiltonians, and that a similarity transform of the function
converges to a joint eigenfunction of the latter four difference operators
- …