The qualitative theory of elementary transformations of one- and multichannel quantum systems in the inverse-problem approach. The construction of transformations with given spectral parameters

New results on simple, universal algorithms of spectrum, scattering, Aand decay control are reviewed. The limiting (most informative) cases Aare studied, such as Athe separation in space and in energy of emingly Ainseparable states close in energy and in the Ashape of the moduli of their wave functions, and conversely, decrease of the level spacing Ato the point where the levels become degenerate. The enrichment of the Aset of elementary Atransformations in going to multichannel systems A(to a vector structure of the spectral parameters, the levers for Acontrolling coupled channels) is demonstrated. For example, it becomes Apossible to concentrate waves in a single channel by "pumping" them Aout of other channels. It becomes clear that soliton-like potential Awells are the "carriers" of not only one-channel, but also partial Achannel states when they are separated from the other states. A Areflectionless interaction which has no analog in the one-channel case Ais described. The "paradoxical" coexistence of confinement and Atransparency at the same energy for the same interaction is discussed. A© 7999 American Institute of Physics. [SI 063-7796(99)00102-3]