Learning Coulomb Potentials and Beyond with Free Fermions in Continuous Space

The first-principles formulation of quantum mechanics relevant for quantum chemistry and trapped quantum gases involves particles in the continuous space $\mathbb R^d$. We present a unified framework and modular algorithm for learning external potentials $V$ with free-fermion models in the continuum. Compared to the lattice-based approaches, the continuum presents new mathematical challenges: the state space is infinite-dimensional and the Hamiltonian contains the Laplacian, which is unbounded in the continuum and produces an unbounded speed of information propagation. We address these through novel optimization methods and information-propagation bounds in combination with a priori regularity assumptions on the external potential. The resulting algorithm provides a unified and robust approach to learn parametric interactions (e.g., Coulomb potentials or periodic potentials) and general smooth functions. Our results lay the foundation for a scalable and generalizable toolkit to learn Hamiltonians in continuous space.