  A bound state problem in a topologically massive quantum electrodynamics is
investigated by using a non-perturbative method. We formulate the Bethe-
Salpeter equation for scalar bound states composed of massive fermion and
anti-fermion pair under the lowest ladder approximation. In a large mass
expansion for the (anti-) fermion, we derive the Schr{\"o}dinger equation and
solve it by a numerical method. The energy eigenvalues of bound states are
evaluated for various values of a topological mass and also a fermion mass.
Then we find a novel logarithmic scaling behaviour of the binding energy in
varying the topological mass, fermion mass and also a quantum number. There
exists a critical value of the topological mass, beyond which the bound states
disappear. As the topological mass decreases, the energy eigenvalues of the
bound states, which are negative, also decrease with a logarithmic dependence
on the topological mass. A Chern-Simons term gives the bound system a repulsive
effect.
