Asymptotics of Eigenvalues for Differential Operators of Fractional Order

TL;DR

This paper analyzes the asymptotic behavior of eigenvalues for a class of multidimensional fractional differential operators, providing classification criteria and asymptotic formulas for their spectral properties.

Contribution

It introduces a classification of fractional differential operators based on Schatten-von Neumann class membership and derives an asymptotic formula for their eigenvalues.

Findings

Operators are classified by resolvent Schatten-von Neumann class

Sufficient conditions for completeness of root functions are formulated

Asymptotic eigenvalue formulas are obtained

Abstract

In this paper we deal with a second order multidimensional fractional differential operator. We consider a case where the leading term represented by the uniformly elliptic operator and the final term is the Kipriyanov operator of fractional differentiation. We conduct classification of such a type of operators by belonging of their resolvent to the Schatten-von Neumann class and formulate the sufficient condition for completeness of the root functions system. Finally we obtain an asymptotic formula.