On Perturbation of Operators and Rayleigh-Schrödinger Coefficients

Let A and E be self-adjoint matrices or operators on ²(N) ℓ 2 ( N ) , where A is fixed and E is a small perturbation. We study how the eigenvalues of $$A+E$$ A + E depend on E , with the aim of obtaining second order formulas that are explicitly computable in terms of the spectral decomposition of A and a certain block decomposition of E . In particular we extend the classical Rayleigh-Schrödinger formulas for the one-parameter perturbation $$A+tE$$ A + t E where t∈ R t ∈ R varies and E is held fixed, by dropping t and considering E as the variable.