Standard eigensolvers for differential operators (or large matrices) are designed to return eigenpairs for which the eigenvalues are constrained in some way: the smallest six eigenvalues (in magnitude), the six eigenvalues nearest a given target, or all of the eigenvalues lying inside a given complex contour, for example. In some cases, however, one would rather select eigenpairs based desired features of the eigenvectors. For example, one may wish to identify eigenpairs for which the eigenvectors are strongly spatially localized. Anderson localization, first posited in the late 1950s, is an important example of this, and has generated an enormous amount of research in the intervening years, with significant contributions from the Simons Collaboration on Localization of Waves in the past decade.

In the first part of this talk, we describe an algorithm for computing eigenpairs for which the eigenvectors are ``sufficiently close'' to a user-specified subspace. We provide heuristic, theoretical and empirical support for this approach. We focus on the magnetic Schrödinger operator, though many of the results carry over naturally to other selfadjoint operators. The second part of the talk concerns a relatively simple modification of the operator---a (canonical) gauge transform---that enables more efficient approximation of eigenpairs.

Fractional diffusion on bounded domains is notorious for the lack of boundary regularity of solutions regardless of the smoothness of domain boundary. We explore this matter for the homogeneous Dirichlet problem for fractional-order quasi-linear operators with variable coefficients in Lipschitz domains and any dimensions; this includes fractional p-Laplacians and operators with finite horizon. We prove lift theorems in Besov norms which are consistent with the boundary behavior of solutions in smooth domains. The proof exploits the underlying variational structure and uses a new and flexible local translation operator. We further apply these regularity estimates to derive novel error estimates for finite element approximations of fractional p-Laplacians and present several simulations that reveal the boundary behavior of solutions.