Certain eigenvalue problems involving the invariant Laplacian on moduli spaces have potential applications to scattering problems in physics. Green, Russo, Vanhove, et al., discovered the behavior of gravitons (hypothetical particles of gravity represented by massless string states) is also closely related to eigenvalue problems for the Laplace-Beltrami operator on various moduli spaces. In this talk we will examine applications and results related to solutions $(\Delta - \lambda) f = E_aE_b$ on $SL_2(\mathbb{Z})\backslash SL_2(\mathbb{R})/SO_2(\mathbb{R})$, where $E_s$ is a non-holomorphic Eisenstein series on $GL(2)$ and $\Delta = y^2(\partial_x^2+\partial_y^2)$. One such interesting finding from this work is a family of identities relating convolution sums of divisor functions to Fourier coefficients on modular forms. This work is in collaboration with Ksenia Fedosova, Stephen D. Miller, Danylo Radchenko, and Don Zagier.