# Introducing a Power of the Operator in Direct Spectral Problems

G.A. Zakirova, E.V. KirillovThe 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.Full text

- Keywords
- regularized trace; Laplace operator; power of operator.
- References
- 1. Zakharov V.E., Faddeev L.D. Korteweg-de Vries Equation: A Completely Integrable Hamiltonian System. Functional Analysis and its Applications, 1971, vol. 5, no. 4, pp. 280-287. DOI: 10.1007/BF01086739

2. Lifshits I.M. [On a Problem of Perturbation Theory Connected with Quantum Statistics]. Uspekhi matematicheskikh nauk [Russian Mathematical Surveys], 1952, vol. 7, no. 1, pp. 171-180. (in Russian)

3. Gel'fand I.M., Levitan B.M. [An Identity for the Eigenvalues of Second Order Differential Operator]. Doklady Akademii Nauk SSSR, 1991, vol. 84, no. 4, pp. 593-596. (in Russian)

4. Levitan B.M., Gasymov M.G. [Determination of a Differential Equation by two of its Spectra]. Russian Mathematical Surveys, 1964, vol. 19, no. 2, pp. 1-63. DOI: 10.1070/RM1964v019n02ABEH001145.

5. Torshina O.A. [The Formula for the First Regularized Trace for the Laplace-Beltrami Operator with Nonsmooth Potential in the Projective Plane]. Differentsial'nye uravneniya i ikh prilozheniya [Differential Equations and Their Applications], Samara, 2006, pp. 32-40. (in Russian)

6. Zakirova G.A., Sedov A.I. [Asymptotics of the Eigenvalues of the Chebyshev Type Operator with Complex Occurrence of the Parameter]. tVestnik Magnitogorskogo gosudarstvennogo universiteta [Bulletin of Magnitogorsk State University], 2004, issue 6, pp. 65-73. (in Russian)

7. Sedov A.I. [On the Existence of Solutions of the Inverse Problem of Spectral Analysis for Self-adjoint Operator in a Hilbert Space]. Obozrenie prikladnoy i promyshlennoy matematiki [Review of Applied and Industrial Mathematics], 2010, vol. 17, issue 3, pp. 454-455. (in Russian)

8. Dubrovskiy V.V., Nagornyy A.V. [ Stability of the Solution of Inverse Problems]. Differentsial'nyy uravneniya [Differential Equations], 1992, vol. 28, no. 5, pp. 839-843. (in Russian)

9. Zakirova G.A. Obratnye spektral'nye zadachi dlya operatora Laplasa s kratnym spektrom. Priblizhennoe vosstanovlenie potentsiala [Inverse Spectral Problem for the Laplace Operator with Multiple Spectrum. Approximate Recovery Potential]. Saarbrucken, LAPLAMBERT Academic Publishing, 2011. (in Russian)

10. Titchmarsh E.C. Eigenfunction Expansion Associated with Second Order Differential Equations. Oxford At The Clarendon Press, 1961.