= Hyperbolic Laplacian
{title2=$\Delta_{\mathbb H^2}=-y^2(\partial_x^2+\partial_y^2)$}
The <hyperbolic Laplacian> is the <Laplace-Beltrami operator> for the metric of <hyperbolic space>. In upper-half-space coordinates $(x_1,\ldots,x_{n-1},y)$ with $y>0$,
$$
ds^2=\frac{dx_1^2+\cdots+dx_{n-1}^2+dy^2}{y^2}.
$$
Using the nonnegative convention $\Delta=-\operatorname{div}\operatorname{grad}$ gives
$$
\Delta_{\mathbb H^n}=-y^2\left(\sum_{j=1}^{n-1}\partial_{x_j}^2+\partial_y^2\right)+(n-2)y\partial_y.
$$
Indeed, insert $g^{ij}=y^2\delta^{ij}$ and $\sqrt{\det g}=y^{-n}$ into the coordinate formula $-\left(\det g\right)^{-1/2}\partial_i(\sqrt{\det g}\,g^{ij}\partial_j)$. The choice of overall sign must be specified when comparing spectral formulas.<Analysis on Manifolds via the Laplacian by Yaiza Canzani>[section 4.1, Definition 14 and the following local-coordinate formula, printed page 43]{ref}
In dimension two the first-derivative term disappears. For $f(y)=y^s$, direct differentiation gives $\Delta f=s(n-1-s)f$, and in particular $s(1-s)f$ in the <hyperbolic plane>. This is the eigenvalue expression underlying many <Maass form> conventions.
Despite its name, this is an <elliptic differential operator> on $y>0$: its leading symbol in the chosen sign convention is $y^2|\xi|^2$. The adjective hyperbolic refers to the geometry, not to the classification of a <hyperbolic partial differential equation>.
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