On the Spectrum of Degenerate Operator Equations

We study the distribution in the complex plane $\mathbb C$ of the spectrum of the operator $L=L(\alpha,a,A)$, $\alpha\in\mathbb R$, $a\in\mathbb C$, generated by the closure in $H=\mathscr L_2(0,b)\otimes\mathfrak H$ of the operation $t^\alpha aD_t^2+A$ originally defined on smooth functions $u(t)\colon[0,b]\to\mathfrak H$ with values in a Hilbert space $\mathfrak H$ satisfying the Dirichlet conditions $u(0)=u(b)=0$. Here $D_t\equiv d/dt$ and $A$ is a model operator acting in $\mathfrak H$. Criterial conditions on the parameter $\alpha$ for the eigenfunctions of the operator $L\colon H\to H$ to form a complete and minimal system as well as a Riesz basis in the Hilbert space $H$ are given.