Laboratory of Information technologies,

Joint Institute for Nuclear Research,

141980 Dubna, Moscow reg.,

RUSSIA

and

Joint Institute of Power Engin. Nuclear Research,

Minsk,

BELARUS

E-mail: suzko@jinr.ru

**Darboux transformations for the matrix Schr\"odinger
equation**

**Abstract:**

The intertwining operator technique is applied to the matrix Schr\"odinger
equation. The first- and second-order matrix Darboux transformations, factorization,
supersymmetry, chains of transformations are studied. A relation between
the matrix first-order Darboux transformation and supersymmetry is considered.
The main differences between the matrix supersymmetry and the standard
scalar supersymmetry for one Schr\"odinger equation and for the coupled
systems of equations are

discussed. An interrelation is established between the differential
and integral transformation operators. It is shown that in a particular
case the second-order integral transformations turn into expressions
for matrix solutions and potentials obtained by the inverse problem
with degenerate integral kernels. Using unitary time-dependent transformations,
we construct exactly soluble time-dependent generalizations of exactly
soluble time-independent equations. The approach opens new opportunities
for modelling the quantum dynamic systems with desired properties, for
instance, quantum wells with the properties of dynamic localization.