On Perturbation of Operators and Rayleigh-Schrödinger Coefficients

Let A and E be self-adjoint matrices or operators on \(\ell ^2({{\mathbb {N}}})\), where A is fixed and E is a small perturbation. We study how the eigenvalues of \(A+E\) 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 \(A+tE\) where \(t\in {{\mathbb {R}}}\) varies and E is held fixed, by dropping t and considering E as the variable.