# Reconciliation of the theory of the operator bundles in spectral problems with the strong internal dissipation of an energy.

**Андронова О. А. Reconciliation of the theory of the operator bundles in spectral problems with the strong internal dissipation of an energy. // Taurida Journal of Computer Science Theory and Mathematics, – 2017. – T.16. – №4. – P. 40-**

https://doi.org/10.37279/1729-3901-2017-16-4-40-50

https://doi.org/10.37279/1729-3901-2017-16-4-40-50

We consider the following spectral problem:

$ \lambda^2u − \lambda \beta Ku − {\Delta} u = 0 (в\;\Omega), \frac{\delta u}{\delta n} + u = 0 (на\;Γ), K = K^∗ \gg 0 $ (1)

Here $\Omega \subset R^m$ is an domain with Lipschitz boundary $Γ = \delta \Omega$. The parameter $\beta > 0$ imitates the power of the internal dissipation of an energy.

The problem (1) can be reduce to study another spectral problem seeing in sum of Hilbert spaces:

$\begin{pmatrix} \beta K & iA^{1/2}\\ iA^{1/2} & 0\\ \end{pmatrix} \begin{pmatrix} u\\ \zeta\\ \end{pmatrix} = \lambda \begin{pmatrix}u\\ \zeta \\ \end{pmatrix}, u \in \mathscr{D}(A) \cap \mathscr{D}(K), \zeta \in \mathscr{D}(A^{1/2}), A \gg 0$ (2)

We consider here, that $0 < A^{−1} \in \mathscr{S}_\infty(E)$.

This problem contains parameter of internal dissipation of an energy $\beta > 0$. It is found out that behavior of spectrum depends on intensity of internal dissipation in the system. It can be weak, middle and strong. The case of strong intensity of internal dissipation is studied in the article. The aim of consideration of this problem is a desire to trace, as spectrum mutates at different positive $\beta$ and to obtain the statements about localization of the spectrum and the properties of own and joined elements. The two methods of the spectral theory of the operator bundles and the theory of the self-adjoint operators in idefinite metric spaces can be used. The first one give that the spectrum has two branches of positive eigenvalues with limit points not only in infinity, but also in zero. Eigenfunctions answering to each branch in the case of strong intensity of internal dissipation in all range of $\beta$ form basis Rissa in some Hilbert spaces.

**Keywords:** Hilbert space, compact self-adjoint operator, classes of compact operators, characteristic equation, dynamics of the eigenvalues motion.