Kernel functions and B"acklund transformations for relativistic Calogero-Moser and Toda systems

We obtain kernel functions associated with the quantum relativistic Toda systems, both for the periodic version and for the nonperiodic version with its dual. This involves taking limits of previously known results concerning kernel functions for the elliptic and hyperbolic relativistic Calogero-Moser systems. We show that the special kernel functions at issue admit a limit that yields generating functions of B"acklund transformations for the classical relativistic Calogero-Moser and Toda systems. We also obtain the nonrelativistic counterparts of our results, which tie in with previous results in the literature.

💡 Research Summary

This paper develops explicit kernel functions for the quantum relativistic Toda lattice, covering both the periodic version and the non‑periodic version together with its dual system. The construction is achieved by taking carefully chosen limits of previously known kernel functions for the elliptic and hyperbolic relativistic Calogero‑Moser models.

The authors begin by recalling the elliptic kernel (K_{\mathrm{ell}}(x,y;\lambda)) and the hyperbolic kernel (K_{\mathrm{hyp}}(x,y;\lambda)) that intertwine the corresponding quantum Hamiltonians. By scaling the particle coordinates and the coupling parameters in a coordinated way—essentially sending a small parameter (\varepsilon) to zero while simultaneously rescaling the spectral parameter—they show that both kernels converge to a single function (K_{\mathrm{Toda}}(x,y;\gamma)). This limit produces the kernel for the periodic relativistic Toda chain. For the non‑periodic chain, an additional Fourier‑type transformation introduces dual variables, leading to a dual kernel (K_{\mathrm{Toda}}^{*}(p,q;\gamma)). The convergence proofs rely on detailed asymptotics of multiple gamma and beta functions, ensuring uniform control over the complex plane and preserving the required analytic properties.

A central result is that the logarithm of these kernel functions, in the classical limit (\hbar\to0), becomes a generating function for a Bäcklund transformation of the corresponding classical relativistic Calogero‑Moser or Toda system. Explicitly, if one defines
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