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Joint eigenfunctions for the relativistic Calogero-Moser Hamiltonians of hyperbolic type. I: First steps. (English) Zbl 1325.81097

The authors of this interesting paper investigate the recursion scheme to construct joint eigenfunctions for the commuting analytic difference operators associated with the integrable \(N\)-particle systems of hyperbolic relativistic Calogero-Moser type. The scheme is based on a kernel identities which is obtained in previous works of the same authors. The relativistic generalization (relativistic Hamiltonian) by the commuting analytic difference operators (\(A\triangle O\)s) has the form \[ S_k (x) = \sum\limits_{I\subset \{1\ldots N\},|I|=k}\prod\limits_{m\in I, n\notin I}f_{-}(x_m-x_n) \prod\limits_{l\in I}\exp{(-i\hbar\beta\partial_{x_l})}\prod\limits_{m\in I, n\notin I}f_{+}(x_m-x_n) \] where \(k=1\ldots N\), and \(f_{\pm }(z) = (\sinh{ (\mu (z \pm i\beta g)/2)}/ \sinh{(\mu z/2))^{1/2}}\); \(\beta = 1/mc > 0\) is the classical multiplier, \(c\) - speed of light, \(m\) - mass. The authors focus on a complete proof of the joint eigenfunction properties in the suitable holomorphy domains. The recursive scheme for constructing joint eigenfunctions \(J_N\) of the \(A\triangle O\)s \(A_{k,\delta }\) are investigated. They establish holomorphy domains and uniform decay bounds which are sufficient for proving that “the scheme does provide well-defined functions \(J_N\) that satisfy the expected joint eigenvalue equations”. The present paper is a first one of a series, that the authors present “the formal features of the scheme, show explicitly its arbitrary-\(N\) viability for the free cases, and supply the analytic tools to prove the joint eigenfunction properties in suitable holomorphy domains”.

MSC:

81R12 Groups and algebras in quantum theory and relations with integrable systems
37J35 Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests
39A70 Difference operators
47B39 Linear difference operators
70H40 Relativistic dynamics for problems in Hamiltonian and Lagrangian mechanics