About the directed disturbance of discrete operators

Under the directed perturbed operator we will understand such an operator, which is built from the disturbance of a discrete operator by the operators of shift, in such a way that one of the eigunefunctions of the built operator is any preassigned nonvanishing function from the domain of operator. By solving problems of theoretical function we often use the spectrum of researched operator, and in many cases only its discrete part, but sometimes together with eigen (and attached) functions. Probably for the researcher it is important to know that it is possible by means of disturbing the given operator to receive an operator with resembling on the primary by the spectrum, and preassigned function which is any by number to the eigenfunction of the disturbed operator.