The Fröhlich Polaron at large total momentum in d=3

We consider the three-dimensional Fröhlich polaron at fixed coupling strength $\alpha >0$ and total momentum $P\in {\text{bbR}}^3$. We show that the bottom of the spectrum reaches the essential spectrum at a finite value of $\mathrm{\mid }P\mathrm{\mid }$, implying that the fiber Hamiltonian has no ground state at sufficiently large total momentum. This extends the known weak-coupling result to arbitrary coupling. Our approach describes states near the spectral threshold by a fully interacting polaron with small residual momentum and one additional phonon carrying almost all of the total momentum.