# The Laughlin liquid in an external potential

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## Abstract

We study natural perturbations of the Laughlin state arising from the effects of trapping and disorder. These are *N*-particle wave functions that have the form of a product of Laughlin states and analytic functions of the *N* variables. We derive an upper bound to the ground state energy in a confining external potential, matching exactly a recently derived lower bound in the large *N* limit. Irrespective of the shape of the confining potential, this sharp upper bound can be achieved through a modification of the Laughlin function by suitably arranged quasi-holes.

## Keywords

Fractional quantum hall effect Laughlin states Quasi-holes## Mathematics Subject Classification

81V70## Notes

### Acknowledgements

We thank Elliott H. Lieb for helpful remarks. N. Rougerie received financial support from the French ANR Project ANR-13-JS01-0005-01.

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