[2603.23173] A Schrödinger Eigenfunction Method for Long-Horizon Stochastic Optimal Control

Computer Science > Machine Learning arXiv:2603.23173 (cs) [Submitted on 24 Mar 2026] Title:A Schrödinger Eigenfunction Method for Long-Horizon Stochastic Optimal Control Authors:Louis Claeys, Artur Goldman, Zebang Shen, Niao He View a PDF of the paper titled A Schr\"odinger Eigenfunction Method for Long-Horizon Stochastic Optimal Control, by Louis Claeys and 3 other authors View PDF HTML (experimental) Abstract: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 $\mathcal{L}$. We prove that, under the gradient drift assumption, $\mathcal{L}$ is unitarily equivalent to a Schrödinger operator $\mathcal{S} = -\Delta + \mathcal{V}$ with purely discrete spectrum, allowing the long-horizon control to be efficiently described via the eigensystem of $\mathcal{L}$. This connection provides two key results: first, for a symmetric linear-quadratic regulator (LQR), $\mathcal{S}$ matches the Hamiltonian of a quantum harmonic oscillator, whose closed-form eigensystem yields an analytic solution to the symmetric LQR with \emph{arbitrary} terminal cost. Second, in a more general setting, we learn the eig...