  Interpretation of D1 and D0-branes in 1+1 string theory as classical and
quantum eigen-values in dual c=1 Matrix Quantum Mechanics (MQM) was recently
suggested. MQM is known to be equivalent to a system of N free fermions
(eigen-values). By considering quantum mechanics of fermions in the presence of
classical eigen-value we are able to calculate explicitly the perturbation of
the shape of Fermi-sea due to the interaction with the brane. We see that the
shape of the Fermi-sea depending on the position of the classical eigen-value
can exhibit critical behavior, such as development of cusp. On quantum level we
build explicitly the operator describing quantum eigen-value. This is a vertex
operator in bosonic CFT. Its expectation value between vacuum and Dirichlet
boundary state is equal to the correct wave-function of the fermion. This
supports the conjecture that quantum eigen-value corresponds to D0-brane. We
also show that c=1 MQM can be obtained as analytical continuation of the system
of 2d electrons in magnetic field which is studied in Quantum Hall Effect.
