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.- L. Chen, M. Dreher: Quantum semiconductor models.- H. Bel Hadj Ali, A. Ben Amor, J. Brasche: Large coupling convergence: Overview and new results.- M. Ben-Artzi: Smooth spectral calculus.- W. Bauer, K. Furutani, C. Iwasaki: Spectral analysis and geometry of sub-Laplacian and related Grushin-type operators.- G. Mendoza: Zeta functions of elliptic cone operators.- P. McKeag, Y. Safarov: Pseudodifferential operators on manifolds: A coordinate-free approach.