  We study spectral properties of the Schroedinger operator with an imaginary
sign potential on the real line. By constructing the resolvent kernel, we show
that the pseudospectra of this operator are highly non-trivial, because of a
blow-up of the resolvent at infinity. Furthermore, we derive estimates on the
location of eigenvalues of the operator perturbed by complex potentials. The
overall analysis demonstrates striking differences with respect to the
weak-coupling behaviour of the Laplacian.
