The de Broglie relations connecting the wave-particle dual nature as generalised from photons to all matter particles are given by $$E=\hbar\omega$$ and $$p=\hbar k$$ (Characterizing a single plane wave with wave parameters $\omega$ and $k$ .)
Are these relations true for a time dependent potential $V(x,t)$ And if yes then is there any proof, if not then can we say that de Broglie's relations are hypotheses?