The spectral asymptotics of a quadratic pencil of differential operators
The authors study the distribution function of eigenvalues of the pencil \(L(\lambda)=A+\lambda B-\lambda^2 I\), where \(A\) is the self-adjoint differential operator induced in \(H=L_2(-\infty,+\infty)\) by the differential expression \(y^{(4)}+q(x)y\) with a positive function \(q(x)\), \(B\) is the self-adjoint operator induced in \(H\) by the expression \((i/2)((p(x)y'')' +(p(x)y')'')\), where \(p(x)\) is a twice continuously differentiable positive function. In this case, \(L(\lambda)\) is not a Keldysh pencil. In Section 1 of the paper, the asymptotic behavior of the Green function \(G(x,\xi,\lambda)\) of the pencil \(L(\lambda)\) for \(\lambda=i\mu\) as \(\mu\to+\infty\) is investigated. In Section 2, the Keldysh Tauberian theorem is used to derive an asymptotic formula for the distribution of eigenvalues of \(L(\lambda)\) from the representation of \(G(x,\xi,\lambda)\).
quadratic pencil of differential operators
scientific article; zbMATH DE number 2176520
scientific article