Some asymptotic properties of the polylaplacian operator
The paper deals with the eigenvalue problem \[ (-1)^m\psi^{(2m)} =\lambda \psi \] on the interval \([-1, 1]\) with the boundary conditions \(\psi^{(j)}(\pm 1)=0\) \((0\leq j\leq m -1)\). The author considers the asymptotic behaviour of the eigenvalues as \(m\to \infty\). He derives strikingly simple asymptotically correct formulae for the Green's function, for the minimal eigenvalue, and for the associated eigenvector. The principal feature observed is that, for large \(m\), the solution operators can be well approximated in all Schatten norms by operators of rank 1. He also shows that a fixed lower-order perturbation term has no effect on the asymptotic behaviour of the solution operator.
scientific article; zbMATH DE number 5962657