Tosio Kato's monumental work on perturbation theory of operators was highly motivated by applications to partial differential equations in non homogeneous media. For example, how is the propagation of waves affected by perturbation of its conductivity? The first attempts in the fifties were by functional analysis methods because it is expected that kinetic energy is related to spectrum. But they were not successful. The connection to harmonic analysis questions was made by A. McIntosh in the eighties. A complete solution occurred in the early 2000 and still implies nowadays unsuspected consequences. This talk will present some aspects of this story.