Introducing a power of the operator in direct spectral problems

The resolvent method, proposed by Sadovnichiy and Dubrovsky in the 1990s, is successfully applied in the direct spectral problem to calculate the asymptotics of eigenvalues of the perturbed operator, find formulas for the regularized trace, and recover perturbation. But the application of this method faces difficulties when the resolvent of the unperturbed operator is non-nuclear. Therefore, a number of physical problems could only be considered on the interval. This article describes a justification of the transition to the power of an operator in order to expand the area of possible applications of the resolvent method. Considering the problem of calculating the regularized trace of the Laplace operator on a parallelepiped of arbitrary dimension, we show that for every fixed dimension it is possible to choose the required power of the operator and to calculate the regularized traces. These studies are relevant due to the need to study important applied problems, particularly in hydrodynamics, electronics, elasticity theory, quantum mechanics, and other fields.