On a non-local problem for irregular equations

We study the distribution on the complex plane $\mathbb C$ of the spectrum
$$ \sigma L=P\sigma L\cup C\sigma L\cup R\sigma L $$
of the operator $L$ generated by the closure in $H=\mathscr L_2(T_1,T_2)\otimes\mathfrak H$ of an irregular operation $a(t)D_t+A$ originally defined on the smooth functions $u(t)\colon[T_1,T_2]\to\mathfrak H$ that satisfy the non-local conditions: $\mu\cdot u(T_1)-u(T_2)=0$. Here $a(t)=\sum_{k=1}^2a_k|t|^{\alpha_k}\chi_k(t)$; $a_k\in\mathbb C$, $a_k\ne 0$; $\alpha_k\in\mathbb R$; $\chi_k(t)$ is the characteristic function of the interval with end-points $0,T_k$; $-\infty<T_1<0<T_2<+\infty$; $D_t\equiv d/dt$; $A$ is a model operator acting in a Hilbert space $\mathfrak H$; $\mu\in\overline{\mathbb C}$, $\mu\ne0,\infty$.