On the Lowest by $x$-variable Terms Influence on the Spectral Properties of Dirichlet Problem for the Hyperbolic Systems

O. V. Alexeeva ; V. V. Kornienko ; D. V. Kornienko

Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences, no. 2 (2014), pp. 16-21

Résumé

We made the comparison study and characterize the spectral properties of differential operators induced by the Dirichlet problem for the hyperbolic system without the lowest terms of the form $$ \cfrac{\partial^2{u^1}}{\partial{t}^2}+\cfrac{\partial^2{u^2}}{\partial{x}^2} = \lambda{u^1}+f^1, \quad \cfrac{\partial^2{u^2}}{\partial{t}^2}+\cfrac{\partial^2{u^1}}{\partial{x}^2} = \lambda{u^2}+ f^2, $$ and for the hyperbolic system with the lowest terms of the form $$ \cfrac{\partial^2{u^1}}{\partial{t}^2}+\cfrac{\partial^2{u^2}}{\partial{x}^2}+\cfrac{\partial{u^2}}{\partial{x}} =\lambda{u^1}+f^1, \quad \cfrac{\partial^2{u^2}}{\partial{t}^2}+\cfrac{\partial^2{u^1}}{\partial{x}^2}+\cfrac{\partial{u^1}}{\partial{x}} = \lambda{u^2}+ f^2, $$, which are considered in the closure $V_{t,x}$ of the bounded domain $\Omega_{t,x}=(0;\pi)^2$ in Euclidean space $\mathbb{R}^2_{t,x}.$ The spectral properties of the boundary value problems for the systems of linear differential equations of the hyperbolic type are investigated in the Hilbert space $\mathcal{H}_{t,x}$ in the terms of spectral closed operators $L:\mathcal{H}_{t,x}\to\mathcal{H}_{t,x}$. We study the spectra of the closed differential operators $L:\mathcal{H}_{t,x}\to\mathcal{H}_{t,x},$ induced by the Dirichlet problem for the second order hyperbolic systems: $C\sigma{L}=R\sigma{L}$ — empty set; point spectrum $P\sigma{L}$ is in the real straight line of the complex plane $\mathbb{C}$. The operator $L$ eigen vector functions generate the orthogonal basis for the hyperbolic system without the lowest terms. For the hyperbolic system with the lowest terms the operator $L$ eigen vector functions generate the Riesz basis, nonorthogonal in the Hilbert space $\mathcal{H}_{t,x}.$ The theorems on the structure of the induced by the Dirichlet problem operator $L$ spectrum $\sigma L$ are formulated.

Détail

Keywords: hyperbolic systems, boundary value problems, closed operators, spectrum, basis, orthogonal basis, Riesz basis.