Asymptotics of the spectrum of a singular nonsemibounded fourth-order vector differential operator in a vector function space (Q610556)

This paper deals with the asymptotics of the spectrum of the fourth-order vector minimal differential operator \(L_0\) defined by \(L_0(y)=y^{(4)}+Q(x)y\), \(y=(y_1(x),y_2(x))\), \(0<x<+\infty\), in the function space \(H=L^2(0,+\infty)\oplus L^2(0,+\infty)\). The potential \(Q(x):=\|q_{ij}(x)\|^2_{i,j=1}\) is a real matrix with eigenvalues \(\mu_i(x)\rightarrow -\infty\) as \(|x|\rightarrow +\infty\). The analysis developed in this paper is carried out in terms of the rotation velocity of eigenvectors of a potential matrix. The proofs combine techniques of asymptotic analysis with Carleman's method and the two-sided Tauberian theorem.