Alec Shelley

#### Stanford

### “The Potential Inversion Theorem”

In a tight binding model of a solid, electrons are free to hop around between nearest-neighbor sites and interact with one another through Coulomb repulsion. It is a surprising fact of this model that two electrons placed near each other will be dynamically bound together for all time if their Coulomb repulsion strength is large enough. It turns out that this effect is general for lattice models and is caused directly by the discretization of space.

I proved the potential inversion theorem, which says that for initial conditions occupying only odd or even parity lattice sites, time evolution is preserved by inverting the potential energy landscape, V→ -V. This explains the bound electron pair, and in fact says that two electrons will time-evolve in position space exactly like an electron and a positron. This effect illustrates and relates several seemingly unrelated physical phenomena, including Bloch oscillations, particle-hole symmetry, and negative-potential scattering.

### ABSTRACT

I study differential equations relevant to physics like the Schrodinger equation, and how they change when they are discretized, or noise/disorder are added. Specifically, I found an effect caused by a discrete, crystalline lattice, where two electrons can get stuck together dynamically under time evolution governed by a nearest neighbor hopping model.

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