Generalized eigenfunctions and spectral theory for strongly local Dirichlet forms

We present an introduction to the framework of strongly local Dirichlet forms and discuss connections between the existence of certain generalized eigenfunctions and spectral properties within this framework. The range of applications is illustrated by a list of examples

A guide to the Choquard equation

We survey old and recent results dealing with the existence and properties of solutions to the Choquard type equations $$-\Delta u + V(x)u = \bigl(|x|^{-(N-\alpha)} * |u|^p\bigr)|u|^{p - 2} u \qquad \text{in $\mathbb{R}^N$},$$ and some of its variants and extensions.Comment: 39 page