The rigorous study of electronic behavior at the fundamental scale relies on solving the Schrödinger equation. While this is commonly approached using differential methods, an equally powerful and often more advantageous strategy is the use of an integral formulation, transforming the partial differential equation into a Boundary Integral Equation (BIE) or Volume Integral Equation (VIE). These formulations naturally enforce boundary conditions and are especially powerful for systems confined to specific domains, requiring discretization of only said domains or their boundaries. However, the resulting integral equations lead to dense matrix systems due to the non-local nature of integral operators. For realistic systems with millions of degrees of freedom, the computational cost scales unfavorably, severely limiting the size and complexity of problems that can be addressed.