Perturbation theory of multiparameter eigenvalue problems (Q1103165)

Let \(T_ r\) and \(V_{rs}\) be linear selfadjoint operators in Hilbert spaces \({\mathfrak H}_ r\). Let \(\lambda_ s\) be complex parameters. Then it is considered the multiparameter system \[ (T_ r+\sum^{k}_{s=1}\lambda_ sV_{rs})u_ r=0,\quad r=1,...,k. \] The main objective is to study the influence of perturbations for the operators \(T_ r\). It is assumed that the perturbed operators \(T_ r(\theta)\) depend holomorphically on a parameter \(\theta\). The behaviour of the eigenvalues and eigenvectors as functions of \(\theta\) is investigated.