 Hyperbolic Laplacian

The hyperbolic Laplacian is the Laplace-Beltrami operator for the metric of hyperbolic space. In upper-half-space coordinates $(x_1,\dots ,x_{n-1},y)$ with $y>0$,Using the nonnegative convention $\Delta =-\mathrm{div}\mathrm{grad}$ givesIndeed, 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.^[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.