  Normal modes, quasi-normal modes and super-radiant modes are studied to
clarify the total dynamics for complex scalar fields in Kerr anti-de Sitter
black hole spacetime. Orthonormal relations and quasi-orthonormal relations are
obtained for normal modes and quasi-normal modes. Mode expansions are done and
the conserved quantities are studied. Any modes are shown to be separated into
two groups, physical modes and unphysical modes, by the zero mode line. Zero
modes themselves do not exist as normalizable modes with the correct boundary
condition. The allowed physical modes exclude the super-radiant instability
modes in rotating black hole spacetime. The result is consistent with the
co-rotating frame consideration.
