  One of the most important issues of quantum engineering is the construction
of low-dimensional structures possessing desirable properties. For example, in
different areas of possible applications of the structures containing quantum
wells (QW), there is need to have QW energy spectrum, which is predetermined.
  Then the following question arises: can one reconstruct the shape of QW which
supports this spectrum? We outline the possible strategy of the QW potential
shape reconstruction, if the spectrum of QW is given in advance. The proposed
approach is based on the combination of different techniques such as Inverse
Scattering Problem Method, Darboux and Liouville transformation. It enables to
take into account the space-variable dependent effective mass of charge
carriers and allows the kinetic energy operator to be of non-Hermitian as well
as Hermitian form. The proposed technique allows to construct phase-equivalent
potentials, to add the new bounded states to (or remove some of them from) the
spectrum supported by an initial potential and provides a systematic procedure
for generating new exactly solvable models.
