One-parameter discrete-time Calogero–Moser system

Abstract

We present a new type of integrable one-dimensional many-body systems called a one-parameter Calogero–Moser system. At the discrete level, the Lax pairs with a parameter are introduced and the discrete-time equations of motion are obtained as together with the corresponding discrete-time Lagrangian. The integrability property of this new system can be expressed in terms of the discrete Lagrangian closure relation by using a connection with the temporal Lax matrices of the discrete-time Ruijsenaars–Schneider system, an exact solution, and the existence of a classical $r$-matrix. As the parameter tends to zero, the standard Calogero–Moser system is recovered in both discrete-time and continuous-time forms.

Appendix: The connection between the Lagrangian and the $$\mathbf M_{\mathrm{RS}}$$ matrix of the RS model

In this appendix, we derive the connection between the one-parameter discrete-time Lagrangian and the RS model matrix

$$\begin{aligned} \, \mathbf M_{\mathrm{RS}}=\sum_{i,j=1}^N\frac{\tilde h_ih_j}{\tilde x_i-x_j+\lambda}E_{ij}. \end{aligned}$$

(A.1)

For simplicity, we start with the case of a $2\times 2$ matrix given by

$$\mathbf M_{\mathrm{RS}}= \begin{bmatrix} \dfrac{\tilde h_1h_1}{\tilde x_1-x_1+\lambda} & \dfrac{\tilde h_1h_2}{\tilde x_1-x_2+\lambda} \\ \dfrac{\tilde h_2h_1}{\tilde x_2-x_1+\lambda} & \dfrac{\tilde h_2h_2}{\tilde x_2-x_2+\lambda} \end{bmatrix}.$$

We then compute the determinant

$$\begin{aligned} \, \det\mathbf M_{\mathrm{RS}}&= \frac{\tilde h_1h_1\tilde h_2h_2}{(\tilde x_1-x_1+\lambda)(\tilde x_2-x_2+\lambda)}- \frac{\tilde h_2h_1\tilde h_1h_2}{(\tilde x_2-x_1+\lambda)(\tilde x_1-x_2+\lambda)}= \\ &=h_1\tilde h_1h_2\tilde h_2 \biggl[\frac{1}{(\tilde x_1-x_1+\lambda)(\tilde x_2-x_2+\lambda)}-\frac{1}{(\tilde x_2-x_1+\lambda)(\tilde x_1-x_2+\lambda)}\biggr]= \\ &=h_1\tilde h_1h_2\tilde h_2 \biggl[\frac{(\tilde x_2-x_1+\lambda)(\tilde x_1-x_2+\lambda)- (\tilde x_1-x_1+\lambda)(\tilde x_2-x_2+\lambda)}{\prod_{i,j=1,2} (\tilde x_i-x_j+\lambda)}\biggr]. \end{aligned}$$

This equation can be further simplified as follows:

$$ \det\mathbf M_{\mathrm{RS}}=h_1\tilde h_1h_2\tilde h_2 \biggl[\frac{(\tilde x_2-\tilde x_1)(x_1-x_2)}{\prod_{i,j=1,2} (\tilde x_i-x_j+\lambda)}\biggr].$$

(A.2)

Recalling relations [13], we have

$$\begin{aligned} \, &h^2_i=-\frac{\prod_{j=1}^N (x_i-x_j+\lambda)(x_i-\tilde x_j-\lambda)} {\prod_{i,j=1,j\neq i}^N(x_i-x_j)\prod_{j=1}^N (x_i-\tilde x_j)}, \\ &\tilde h_i^2=-\frac{\prod_{j=1}^N (\tilde x_i-x_j+\lambda)(\tilde x_i-\tilde x_j-\lambda)} {\prod_{i,j=1,j\neq i}^N(\tilde x_i-\tilde x_j)\prod_{j=1}^N (\tilde x_i-x_j)}, \end{aligned}$$

and therefore, for $i,j=1,2$,

$$\begin{aligned} \, &h^2_1=-\frac{(x_1-x_1+\lambda)(x_1-x_2+\lambda)(x_1-\tilde x_1-\lambda)(x_1-\tilde x_2-\lambda)}{(x_1-x_2)(x_1-\tilde x_1)(x_1-\tilde x_2)}, \\ &\tilde h_1^2=\frac{(\tilde x_1-x_1+\lambda)(\hat x_1-x_2+\lambda)(\tilde x_1-\tilde x_1-\lambda)(\tilde x_1-\tilde x_2-\lambda)} {(\tilde x_1-\tilde x_2)(\tilde x_1-x_1)(\tilde x_1-x_2)}, \\ &h^2_2=-\frac{(x_2-x_1+\lambda)(x_2-x_2+\lambda)(x_2-\tilde x_1-\lambda)(x_2-\tilde x_2-\lambda)} {(x_2-x_1)(x_2-\tilde x_1)(x_2-\tilde x_2)}, \\ &\tilde h_2^2=\frac{(\tilde x_2-x_1+\lambda)(\hat x_2-x_2+\lambda)(\tilde x_2-\tilde x_1-\lambda)(\tilde x_2-\tilde x_2-\lambda)} {(\tilde x_2-\tilde x_1)(\tilde x_2-x_1)(\tilde x_2-x_2)}. \end{aligned}$$

Taking the logarithm gives

$$\begin{aligned} \, \ln|h_1|={}&\frac{1}{2}[\ln|\lambda|+\ln|x_1-x_2+\lambda|+\ln|x_1-\tilde x_1-\lambda|+{} \\ &+\ln|x_1-\tilde x_2-\lambda|-\ln|x_1-x_2|-\ln|x_1-\tilde x_1|-\ln|x_1-\tilde x_2, \\ \ln|\tilde h_1|={}&\frac{1}{2}\bigl[\ln|\lambda|+\ln|\tilde x_1-x_1+\lambda|+\ln|\tilde x_1-x_2+\lambda|-{} \\ &-\ln|\tilde x_1-\tilde x_2-\lambda|-\ln|\tilde x_1-\tilde x_2|-\ln|\tilde x_1-x_1|-\ln|\tilde x_1-x_2|], \\ \ln|h_2|={}&\frac{1}{2}[\ln|\lambda|+\ln|x_2-x_1+\lambda|+\ln|x_2-\tilde x_1-\lambda|+{} \\ &+\ln|x_2-\tilde x_2-\lambda|-\ln|x_2-x_1|-\ln|x_2-\tilde x_1|-\ln|x_2-\tilde x_2|], \\ \ln|\tilde h_2|={}&\frac{1}{2}[\ln|\lambda|+\ln|\tilde x_2-x_1+\lambda|+\ln|\tilde x_2-x_2+\lambda|+{} \\ &+\ln|\tilde x_2-\tilde x_1-\lambda|-\ln|\tilde x_2-\tilde x_1|-\ln|\tilde x_2-x_1|-\ln|\tilde x_2-x_2|]. \end{aligned}$$

Hence,

$$\begin{aligned} \, \det\mathbf M_{\mathrm{RS}}={}&\ln|h_1|+\ln|\tilde h_1|+\ln|h_2|+\ln|\tilde h_2|+\ln|\tilde x_2-\tilde x_1|+\ln|x_1-x_2|-\sum_{i,j=1,2}\ln|\tilde x_i-x_j+\lambda|= \\ ={}&2\ln|\lambda|-\sum_{i,j=1,2}\ln|\tilde x_i-x_j+\lambda|= \sum_{i,j=1,2}\ln|x_i-x_j+\lambda|-\sum_{i,j=1,2}\ln|x_i-\tilde x_j|. \end{aligned}$$

Obviously, for $N$ particles or an $N\times N$ matrix, we have

$$ \det\mathbf M_{\mathrm{RS}}=\sum_{i,j=1}^N\ln|x_i-x_j+\lambda|-\sum_{i,j=1}^N\ln|x_i-\tilde x_j|,$$

(A.3)

which is indeed the discrete-time Lagrangian for the one-parameter CM system.