A remark on eigenvalue perturbation theory at vanishing isolation distance

摘要

Let T be a self-adjoint operator in a separable Hilbert space X, admitting compact resolvent and simple eigenvalues with possibly vanishing isolation distance, and let V be symmetric and bounded. Consider the self-adjoint operator family T(g) : g ∈ ℝ in X defined by T + gV on D(T). A simple criterion is formulated ensuring, for any eigenvalue of T(g), the existence to all orders of its perturbation expansion and its asymptotic nature near g = 0, with estimates independent of the eigenvalue index. An application to a class of Schrödinger operators is described.