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

Reconstruction of Quantum Well Potentials via the Intertwining Operator Technique