Some notes about scanning probe microscopy, nanoengineering and methods of quantum mechanics

In the presented paper, a research program for new vision of nanoengineering has been suggested. In the framework of this program, growth of a solid state surface under the influence of external source of sputtering particles is considered as the problem of optimal control by distributed parameter system, the source being a control. As one of cornerstones in this approach, general theory of the one-dimensional KPZ-equation in approximation of small angles under the action of the spatially-inhomogeneous source has been developed. The complete mathematical model of this process with the Dirac delta function source of external particles illustrates this theory, Green's function for this system having been calculated exactly. Scheme of physical realization of delta-functional source by means of series of carbon nanotubes has been discussed. Opportunity of combined characterization of solid state surface by both atomic force microscopy and scattering data for electromagnetic waves has been demonstrated. The height of the sample has been constructed with help of the Cole-Hopf substitution from single eigenfunction of discrete spectrum being an example of the surface shape under investigation.