The Asymptotic Behaviour of the Sum of Negative Eigenvalues of a Self-Adjoint Operator Given in Semi-Axis

Abstract

In this work, we find the asymptotic formulas for the sum of the negative eigenvalues smaller than $-\varepsilon$ $(\varepsilon >0)$ of a self-adjoint operator $L$ which is defined by the following differential expression $$\ell(y)=-(p(x)y'(x))'-Q(x)y(x)$$ with the boundary condition $y(0) = 0$ in the space in the space $L_{2}(0,\infty ;H)$.