The complex geometry of the free particle, and its perturbations

The Hamiltonian operator describing a quantum particle on a path often extends holomorphically to a complex neighborhood of the path. When it does, it can be seen as the local expression of a complex projective structure, and its perturbations become deformations of that geometric structure. We'll describe the Hamiltonian of a free particle as a complex projective surface, and we'll use tools from quasiconformal geometry to study its perturbations. Our main results are loosely modeled on the algebraic "transformation theory" results that underpin the exact WKB method. They're meant to serve as a foundation for efforts to gain a more geometric understanding of the exact WKB method.