I am studying a recent paper in which the author worked on the rectangular, flat 3 tori. It can be realized, the author explained, as $\mathbb{R}^3 \over (L_1 \mathbb Z \times L_2 \mathbb Z \times L_3 \mathbb Z)$ with $L_j \in (0, \infty),j=1,2,3.$ For notational convenience, we use the coordinates for the standard torus $\mathbb{T}^3 := {\mathbb{R}^3 \over \mathbb{Z}^3}$ and incorporate the geometry of the torus into the Riemannian metric, using the corresponding Laplace-Beltrami operator

$$\triangle = \sum_{j=1}^3 L_j^{-2} \frac{\partial^2}{\partial x_2^2}$$

We then define the Schrodinger propagator $e^{i t \triangle} $ by

$$\mathcal{F}(e^{i t \triangle } f)(\xi)=e^{- 2 \pi i t \sum_{j=1}^3 L_j^{-2} \xi_j^2} \hat{f}(\xi), \,\,\, for\,\,\, \xi =(\xi_1,\xi_2,\xi_3)\in \mathbb{Z}^3.$$

I have some difficulties understanding how he scaled the coordinates of the Laplace operator. Also, I can not get how he got the semigroup. Could you please explain for me in details. Thanks in advance.

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