Asymptotic behavior of the spectra of second-order non-self-adjoint systems of differential operators
The differential operator \((Py)(t)=-(t^ \alpha A(t) y'(t))'+C(t) y(t)\) with \(\alpha \in [0,2)\) and matrix coefficients \(A(t) \subset C^ \infty ([0,1],\text{End} \mathbb{C}^ n)\), \(C(t) \in C([0,1],\text{End} \mathbb{C}^ n)\) is considered on \(L_ 2(0,1)^ n\). The domain of the operator is the space \({\overset \circ H}{_ \alpha^ n}\), where \({\overset \circ H}_ \alpha\) is the closure of \(C^ \infty_ n(0,1)\) with respect to the norm \[ \| w \|_{2,\alpha}=\left( \int^ t_ 0 | t^{\alpha/2} w'(t) |^ 2dt+\int^ t_ 0 | w(t) |^ 2dt \right)^{1/2}. \] The author proves an asymptotic formula for the distribution of the eigenvalues in a certain sector \(\Phi=\{z \in \mathbb{C}:| \arg z |\leq\varphi\}\), \(0<\varphi<\pi\), where it is assumed that the matrix \(A\) has simple eigenvalues lying on the positive real axis and outside \(\Phi\). The positive eigenvalues are denoted by \(p_ 1(t)<\dots<p_ \nu(t)\). The eigenvalues of \(P\) lying in \(\Phi\) are denoted by \(\lambda_ 1,\lambda_ 2,\dots,\) counted according to their multiplicity and in nondecreasing order of their absolute values. Let \(N(\lambda)=\text{card} \{j:| \lambda_ j | \leq \lambda\}\) be the counting function. The main result is: There exist numbers \(M>0\), \(l \in \mathbb{N}\), such that \(\text{Re} \lambda_ k>0\) and \(| \text{Im} \lambda_ k | \leq M (\text{Re} z)^{1/2}\) for all \(k \geq l\) and \(N(\lambda)=c \sqrt \lambda+O(\lambda^{1/3} \ln \lambda)\), \(\lambda \to+\infty\), where \[ c={1 \over \pi} \sum^ \nu_{i=1} \int^ 1_ 0t^{-\alpha/2} p_ i^{-1/2} (t)dt. \] {}.
scientific article; zbMATH DE number 484277