Perturbation Theory: Feshbach-Schur Method

Abstract

As we have seen, many basic questions of quantum dynamics can be reduced to finding and characterizing the spectrum of the appropriate Schrödinger operator. Though this task, known as spectral analysis, is much simpler than the task of analyzing the dynamics directly, it is far from trivial. The problem can be greatly simplified if the Schrödinger operator H under consideration is close to an operator H 0 whose spectrum we already know. In other words, the operator H is of the form H = H κ , where H κ = H 0 + κW, (11.1) H 0 is an operator at least part of whose spectrum is well understood, κ is a small real parameter called the coupling constant, and W is an operator, called the perturbation. (All the operators here are assumed to be self-adjoint.) If the operator W is bounded relative to H 0 , say in the sense that D(H 0) ⊂ D(W) and W u ≤ cH 0 u + c