The fractional quantum Hall effect is one of the most intriguing phenomena of condensed
matter physics, made manifest by the transport properties of 2D electron gases in high
magnetic fields. Laughlin's variational wave function, proposed to approximate the
ground state of such a system, forms the basis of much of our current understanding
of this phenomenon but many of its fundamental properties are poorly understood yet
from a mathematical point of view. Laughlin's wave function describes a highly
correlated quantum fluid, and it is of particular interest to understand the
robustness of its built-in correlations in different settings.
In this talk we will investigate a model for the response of the Laughlin state
to variations of an external potential. This leads to a family of variational
problems of a new type. Our main results are rigorous energy estimates demonstrating a
strong rigidity of the response of the Laughlin state to the external potential.
Joint work with Jakob Yngvason