A Schr"odinger Eigenfunction Method for Long-Horizon Stochastic Optimal Control

High-dimensional stochastic optimal control (SOC) becomes harder with longer planning horizons: existing methods scale linearly in the horizon T, with performance often deteriorating exponentially. We overcome these limitations for a subclass of linearly-solvable SOC problems-those whose uncontrolled drift is the gradient of a potential. In this setting, the Hamilton-Jacobi-Bellman equation reduces to a linear PDE governed by an operator L. We prove that, under the gradient drift assumption, L is unitarily equivalent to a Schr"odinger operator S = -Delta + V with purely discrete spectrum, allowing the long-horizon control to be efficiently described via the eigensystem of L. This connection provides two key results: first, for a symmetric linear-quadratic regulator (LQR), S matches the Hamiltonian of a quantum harmonic oscillator, whose closed-form eigensystem yields an analytic solution to the symmetric LQR with arbitrary terminal cost. Second, in a more general setting, we learn the eigensystem of L using neural networks. We identify implicit reweighting issues with existing eigenfunction learning losses that degrade performance in control tasks, and propose a novel loss function to mitigate this. We evaluate our method on several long-horizon benchmarks, achieving an order-of-magnitude improvement in control accuracy compared to state-of-the-art methods, while reducing memory usage and runtime complexity from O(Td) to O(d).