Dynamical Properties of Quantum Hall Edge States

We consider the dynamical properties of simple edge states in integer ($\nu = 1$) and fractional ($ \nu = 1/2m+1$) quantum Hall (QH) liquids. The influence of a time-dependent local perturbation on the ground state is investigated. It is shown that the orthogonality catastrophe occurs for the initial and final state overlap $|<i|f>| \sim L^{-{1\over{2\nu}}({\delta\over{\pi}})^2}$ with the phase shift $\delta$. The transition probability for the x-ray problem is also found with the index, dependent on $\nu$. Optical experiments that measure the x-ray response of the QH edge are discussed. We also consider electrons tunneling from one dimensional Fermi liquid into a QH fluid. It is argued that for any filling fraction the tunneling from a Fermi liquid to the QH edge is suppressed at low temperatures and we find the nonlinear $I-V$ characteristics $I\sim V^{1/\nu}$.