Nonlinear eigenvalue problems often depend on a physical parameter, about which one seeks to optimize or analyze. We will describe concrete examples from several collaborations, showing how such problems can emerge from the exact reduction of infinite-dimensional linear problems to finite-dimensional nonlinear ones. Our first case is motivated by a continuum Fibonacci quasicrystal model, where the parameter scales the potential in a Schroedinger equation. As second class of examples comes from mechanical systems, where the parameter can reflect a material property, such as a damping coefficient. Contour integral methods approximate eigenvalues within some specified bounded domain. To generalize such methods to parametric problems, we describe a parametric extension of Keldysh's theorem (work with Balicki and Gugercin). The resulting algorithm reduces the dimension of nonlinear eigenvalue problems while allowing changes to eigenvalue multiplicity as the parameter evolves.
