  We show that Bogoliubov equations in one-dimensional systems with piecewise
constant potentials can be always solved. In particular, we analyze in detail
the case where the condensate wavefunction is a real-valued function, and give
the explicit expressions for wavefunctions of Bogoliubov excitations. By means
of these solutions, we consider transmission and reflection properties of
Bogoliubov excitations for two types of potential, namely, a rectangular
barrier and a potential step. The results yield simple and exact examples of
anomalous tunneling effect and quantum evaporation.
