We present a numerically exact study of polaron with quadratic coupling to the oscillator displacement, or ${X}^{2}$ polaron, using two alternative methodological developments. Our results cover both antiadiabatic and adiabatic regimes entire range electron-phonon ${g}_{2}$, from system's stability threshold at attractive ${g}_{2}=\ensuremath{-}1$ arbitrary strong repulsion ${g}_{2}\ensuremath{...
