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Spectral Theory of the Fermi Polaron

  • M. GriesemerEmail author
  • U. Linden
Article
  • 17 Downloads

Abstract

The Fermi polaron refers to a system of free fermions interacting with an impurity particle by means of two-body contact forces. Motivated by the physicists’ approach to this system, the present article describes a general mathematical framework for defining many-body Hamiltonians with two-body contact interactions by means of a renormalization procedure. In the case of the Fermi polaron, the well-known TMS Hamiltonians are shown to emerge. For the Fermi polaron in a box \([0,L]^2\subset \mathbb {R}^2\), a novel variational principle, established within the general framework, links the low-lying eigenvalues of the system to the zero modes of a Birman–Schwinger-type operator. It allows us to show, e.g., that the polaron and molecule energies, computed in the physical literature, are indeed upper bounds to the ground state energy of the system.

Notes

Acknowledgements

We thank Rodolfo Figari, Rafaele Carlone, Alessandro Teta, and Jan Philip Solovej for extended discussions and for the hospitality at the University of Naples, the University La Sapienza in Rome, and the University of Copenhagen. We thank Hans Peter Büchler for bringing the Fermi polaron to our attention and for several inspiring discussions. We thank an anonymous referee for pointing out the work of Andrea Posilicano, and we thank Andrea for helpful remarks about his pertinent publications. Our work was supported by the Deutsche Forschungsgemeinschaft (DFG) through the Research Training Group 1838: Spectral Theory and Dynamics of Quantum Systems.

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Authors and Affiliations

  1. 1.Fachbereich MathematikUniversität StuttgartStuttgartGermany

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