This volume collects six articles on selected topics at the frontier between partial differential equations and spectral theory, written by leading specialists in their respective field. The articles focus on topics that are in the center of attention of current research, with original contributions from the authors. They are written in a clear expository style that makes them accessible to a broader audience. The articles contain a detailed introduction and discuss recent progress, provide additional motivation, and develop the necessary tools. Moreover, the authors share their views on future developments, hypotheses, and unsolved problems.

Preface.- W. Bauer, K. Furutani, C. Iwasaki: Spectral analysis and geometry of a sub-Laplacian and related Grushin type operators.- H. Bel Hadj Ali, A. Ben Amor, J. Brasche: Large coupling convergence: Overview and new results.- M. Ben-Artzi: Smooth spectral theory.- L. Chen, M. Dreher: Quantum semiconductor models.- N. Jacob, A. Potrykus: Some partial differential and pseudodifferential operatos related to random fields.- G. Mendoza: Spectral theory of elliptic cone operators.- Y. Safarov: Approximate spectral projections of the Laplacian on a Riemannian manifold.