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Source identities and kernel functions for deformed (quantum) Ruijsenaars models. (English) Zbl 1303.81071
In this paper the authors derive explicit source identities and kernel functions for a relativistic generalization of models of quantum Calogero-Sutherland systems due to Ruijsenaars. (A function $$F(x,y)$$ is called a kernel function of a pair of Hamiltonian operators $$H(x)$$ and $${\tilde H}(y)$$ if $$(H(x)-{\tilde H}(y)-c)F(x,y) =0$$ for some constant $$c$$. Such identities have been used to find eigenfunctions of the operators $$H,{\tilde H}$$.) The generalization here is in terms of a deformation of analytic difference operators of the form $\left({\mathcal S}^\pm_{\mathcal N}({\mathbf X}:{\mathbf m})-s(ig\beta \sum_{J=1}^{\mathcal N} m_J)/ig\beta s'(0)\right)\Phi({\mathbf X}:{\mathbf m})=0$ where $$\Phi({\mathbf X}:{\mathbf m})=\prod_{1\leq J<K\leq {\mathcal N}}\;\phi(X_J-X_K: m_J,m_k)$$ as in the Ruijsenaars model, but here $$f_\pm(x:m,m')$$ equals either $$(\frac{s(x\pm ig\beta(m+m')/2)}{s(x\pm ig\beta(m-m')/2)})^\frac12$$ or $$1$$, depending on the values of parameters $$m,m'$$. The function $$s(x)$$ is chosen as usual for the rational, trigonometric, hyperbolic and elliptic cases. The Ruijsenaars model is the special case $$m=m'=1$$. The authors find a source identity for these operators of the form $\left({\mathcal S}^\pm_{\mathcal N}({\mathbf X}:{\mathbf m})-s(ig\beta \sum_{J=1}^{\mathcal N} m_J)/ig\beta s'(0)\right)\Phi({\mathbf X}:{\mathbf m})=0$ where $$\Phi({\mathbf X}:{\mathbf m})=\prod_{1\leq J<K\leq {\mathcal N}}\;\phi(X_J-X_K: m_J,m_k)$$ is a common eigenfunction for both $$\pm$$ cases and $$\phi(x:m,m')$$ is either $$s(x)$$ or expressible in terms of a function depending on $$s(x)$$ for the various choices of the parameters. A key observation here is that if $$(m_J,X_J)= (1,x_J)$$ for $$J=1,\dots,N$$ and $$(m_J,X_J)= (-1,y_{J-N})$$ for $$J-N=1,\dots,M$$ then $${\mathcal S}^\pm_{ N+M}({\mathbf X}:{\mathbf m})={\mathcal S}^\pm_{ N}({\mathbf x}:g,\beta) -{\mathcal S}^\pm_{ M}({-\mathbf y}:g,\beta)$$, so that the source identity leads immediately to explicit kernel function identities of the form $\left({\mathcal S}^\pm_{ N}({\mathbf x}:g,\beta) -{\mathcal S}^\pm_{ M}({-\mathbf y}:g,\beta) -s(ig\beta (N-M))/ig\beta s'(0)\right) F_{N,M}({\mathbf x},{\mathbf y}:g,\beta)=0.$ The complete details of the long and complicated proofs are provided; the exposition is clear. There is a discussion of future research directions.

##### MSC:
 81Q05 Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics 16R60 Functional identities (associative rings and algebras) 14D15 Formal methods and deformations in algebraic geometry 81Q80 Special quantum systems, such as solvable systems 30C40 Kernel functions in one complex variable and applications
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