Schrödinger‐Based Path Planning: A Wave‐Diffraction Approach to Global Navigation in Complex Environments

This paper presents a novel approach to global path planning and autonomous navigation utilizing the steady‐state Schrödinger equation. Traditional heuristic search algorithms, such as A ^∗ and RRT ^∗ , often become computationally prohibitive in dynamic environments or when smooth trajectories are required. In contrast, the proposed framework treats the workspace as a continuous physical system governed by a partial differential equation (PDE). By modeling the target configuration as a source term and obstacles as high‐potential barriers within a Hamiltonian operator, the algorithm yields a global wave function Ψ that simultaneously encodes the reachability of every point in the workspace. To overcome numerical instabilities associated with exponential decay and vanishing gradients, we apply the Cole–Hopf transformation, mapping the wave intensity onto a robust potential manifold U . Furthermore, a normalized gradient descent scheme is introduced to maintain a constant traversal velocity and prevent agent stagnation in regions of low wave amplitude. Experimental evaluations in 2D and 3D environments demonstrate that the resulting navigation field generates smooth, C ^∞ ‐continuous, collision‐free trajectories for multiagent systems in near real time. Ultimately, this physics‐informed formulation provides a continuous, smooth, and scalable alternative to discrete graph‐search techniques, offering inherent obstacle avoidance and natural scalability for multirobot navigation.