Perturbation theory of linear operators

Summary

Perturbation theory of linear operators studies how the properties of an operator $T$ change under replacement by $T + V$. Here $V$ is small in a suitable sense. The main questions are whether closedness, self-adjointness, the spectrum and the eigenvalues survive the change, and how they move. The basic notion of smallness for unbounded operators is relative boundedness. The theory underlies much of mathematical quantum mechanics, where a Hamiltonian is a free part plus an interaction.