Finding the condition of finiteness of the discrete spectrum of a nonself-adjoint friedrichs model in the case of one-dimensional perturbation

Abstract

The object of this study is the Friedrichs model in the case of one-dimensional perturbation of the multiplication operator by an independent variable. One of the most problematic places in this theory is when the number of eigenvalues is infinite. Therefore, the important is to find the conditions under which there is a finite number of eigenvalues. In this paper used standard methods of functional analysis, namely: calculation of operator norms, finding of conjugate operator, calculation of functional norms, calculation of operator re-solvent with substantiation of re-solvent existence conditions. Traditionally, the perturbation of the operator is presented in a factorized form (i. e. in the form of the product of two operators, one of which acts from the main space to a certain auxiliary space, and the other, conversely, from the auxiliary space to the main one). In addition to the methods of functional analysis, it is possible to work with improper integrals over an infinite interval. Let’s emphasize that in this paper let’s also use the concepts of smallness by norm and the concept of smallness by dimension. In this case, the dimension of the perturbation operator is one-dimensional. The following statement is obtained: if it is established that the integral has a finite number of eigenvalues and if it is established that the re-solvent tends to zero for σ→∞, then there will be a finite number of eigenvalues on the entire axis. By superimposing a condition on the difference between the perturbation and the conjugate perturbation, let’s find the finiteness of the operator spectrum. Due to the fact that there is a finite range of spectrum, let’s obtain the opportunity to work with expressions on various topics. This fact greatly simplifies all calculations, regardless of the nature of the studied expressions: mechanical, physical or otherwise. Due to the finite number of eigenvalues of the perturbed operator, let’s obtain the advantage that there is no need to sum up an infinite number of terms in expressions because it would actually be impossible.