Renormalised perturbative series for the impurity levels in a quantum well

The energy levels of an impurity center in a deep quantum well of width L and depth g are studied analytically . Renormalised perturbative series are constructed in the regions g L^ << 1 and g L^2 >> 1. Maximal binding energy and wave function deformation to a quasi-twodimensional function are found to occur at a certain L_c satisfying sqrt{g} L_c ~ 1. Similar results may be obtained for the impurity in a quantum wire, in a dot or in a multiwell structure.