Distribution of eigenvalues of nonselfadjoint differential operators of second order (Q2640040)

In the space \(H=L_ 2((0,T);{\mathbb{C}}^ n)\) the differential operator \(Pu=-(t^{\alpha}A(T)u'(t))'+Q(t)u(t),\quad 0\leq \alpha <2,\) is considered with Dirichlet type boundary conditions. A(t) and Q(t) are \(n\times n\) matrices with entries in \(C^{\infty}[0,1]\) and C[0,1], respectively. It is supposed that A(t) is invertible and that its eigenvalues lie on n different rays \(c_ j{\mathbb{R}}^+\) \((j=1,...,n)\). Then it is stated that the eigenvalues of the problem lie in certain asymptotic regions \(S_ j\) around these rays. Let \(N_ j(\lambda)\) denote the number of eigenvalues of the operator P in \(S_ j\) whose absolute values are less than \(\lambda\). Then \(N_ j(\lambda)=c_ j\sqrt{\lambda}+O(1)\lambda^{1/3}\) where \(c_ j=1/\pi \int^{1}_{0}t^{-\alpha /2}| p_ j(t)|^{-1/2}dt\).