Spectral Theory of the Fermi Polaron

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⊂R2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$[0,L]^2\subset \mathbb {R}^2$$\end{document}, 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.