Consider a closed compact genus 0 Riemann surface $\Sigma$ on which we install a metric $g$. The eigenvalue equation for the scalar Laplacian is $$ - \nabla^2 \psi_n(x) = \lambda_n \psi_n(x) , \qquad \int d^2x \sqrt{g} \psi_m \psi_n = V\delta_{mn} , \qquad 0 = \lambda_0 < \lambda_1 \leq \lambda_2 \leq \lambda_3 \leq \cdots $$ The second equation defines our normalization for the eigenfunctions. $V$ is the volume of $\Sigma$. With this normalization, we have $\psi_0(x)=1$. Since $\Sigma$ is closed and compact $\lambda_n$ is a discrete set of eigenvalues which are all positive.
Q: Is there anything known about integrals of the type: $$ \int d^2 x \sqrt{g} \psi_n R , \qquad \int d^2 x \sqrt{g} \psi_m \psi_n R, \qquad \int d^2 x \sqrt{g} \psi_m \psi_n \psi_\ell R , \quad \text{etc.} $$ where $R$ is the Ricci scalar of $(\Sigma,g)$.
For instance, if $g$ is the round sphere metric then $R$ is simply a constant, then these integrals are simply $RV\delta_{n,0}$, $RV\delta_{m,n}$ and $RV C_{mn\ell}$ where $C_{mn\ell}$ are the Clebsch-Gordon coefficients for $SU(2)$.
What can I say, if anything, about these integrals for arbitrary metrics $g$?
NOTE: On a genus 0 Riemann surface, any metric $g$ can always be written in the form $$ g = e^{2\omega(z,{\bar z})} dz d{\bar z} , \qquad \omega(z,{\bar z}) \xrightarrow{|z| \to \infty} - 2 \ln |z| , \qquad z \in {\mathbb C} $$ The first equality is true because any 2D metric is conformally flat. The second constraint is needed to ensure that the metric is well-defined at $z=\infty$ so that we can compactify ${\mathbb C} \to \Sigma$.