Laplace-Beltrami operator (hyperbolic) (reviewed)

The Laplace-Beltrami operator on the hyperbolic upper half-plane is given by $$\Delta = -y^{2} \left(\frac{\partial^{2}}{\partial x^{2}}+\frac{\partial^{2}}{\partial y^{2}}\right).$$ Note that it is sometimes defined without the minus sign in front.