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The Hamiltonian operator describing a quantum particle on a path oftenextends holomorphically to a complex neighborhood of the path. When it does, itcan be seen as the local expression of a complex projective structure, and itsperturbations become deformations of that geometric structure. We'll describethe Hamiltonian of a free particle as a complex projective surface, and we'lluse tools from quasiconformal geometry to study its perturbations. Our mainresults are loosely modeled on the algebraic "transformation theory" resultsthat underpin the exact WKB method. They're meant to serve as a foundation forefforts to gain a more geometric understanding of the exact WKB method.

The complex geometry of the free particle, and its perturbations