I will report on recent results stemming from the analysis of Schrödinger operators on manifolds. I will first describe results dealing with isoperimetric inequalities and optimal (aka extremal) metrics on closed manifolds. These issues have been instrumental in the study of the spectrum of several classical operators, and are motivated by the understanding of the behaviour of the spectrum under changes of metrics. Then, motivated by conjectures of Yau on measures of nodal sets (but which are actually related to the first part of the talk), I will describe how eigenfunctions are concentrating in terms of Lp norms (with an explicit dependence on the eigenvalues). My goal is to emphasize on the case of Schrödinger operators with rough potentials. I will also state several open problems.