Ubiquity of bound states for the strongly coupled polaron

TL;DR

This paper demonstrates that at strong coupling, the polaron's spectrum contains infinitely many excited states, with the number of energy bands diverging as the coupling strength increases.

Contribution

It proves the existence and divergence of excited energy bands for the strongly coupled polaron, using bounds derived from an effective Bogoliubov-type Hamiltonian.

Findings

Number of excited energy bands diverges at strong coupling

Upper bounds for min-max values are established

Comparison with the essential spectrum confirms the results

Abstract

We study the spectrum of the Fr\"ohlich Hamiltonian for the polaron at fixed total momentum. We prove the existence of excited eigenvalues between the ground state energy and the essential spectrum at strong coupling. In fact, our main result shows that the number of excited energy bands diverges in the strong coupling limit. To prove this we derive upper bounds for the min-max values of the corresponding fiber Hamiltonians and compare them with the bottom of the essential spectrum, a lower bound on which was recently obtained in [1]. The upper bounds are given in terms of the ground state energy band shifted by momentum-independent excitation energies determined by an effective Hamiltonian of Bogoliubov-type.