On the lowest by $x$-variable terms influence on the spectral properties of dirichlet problem for the hyperbolic systems

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.

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