Product formulas for the relativistic and nonrelativistic conical functions

TL;DR

This paper derives new product formulas for conical functions and their relativistic counterparts, enabling explicit diagonalizations of related integral operators and exploring their nonrelativistic limits.

Contribution

It introduces novel product formulas for conical and relativistic conical functions, facilitating the explicit diagonalization of commuting integral operators.

Findings

New product formulas for conical functions and their relativistic generalizations.

Explicit diagonalizations of integral operators commuting with 2-particle Hamiltonians.

Uniform limit estimates for hyperbolic gamma functions in nonrelativistic limits.

Abstract

The conical function and its relativistic generalization can be viewed as eigenfunctions of the reduced 2-particle Hamiltonians of the hyperbolic Calogero-Moser system and its relativistic generalization. We prove new product formulas for these functions. As a consequence, we arrive at explicit diagonalizations of integral operators that commute with the 2-particle Hamiltonians and reduced versions thereof. The kernels of the integral operators are expressed as integrals over products of the eigenfunctions and explicit weight functions. The nonrelativistic limits are controlled by invoking novel uniform limit estimates for the hyperbolic gamma function.