The central theme of this workshop is the analysis and computation of Schrödinger operators and applications to nonlinear problems in several areas of Mathematical Physics, Analysis of Partial Differential Equations, Quantum Chemistry, and more. The simplest, most basic example, of such an operator is of the form H = −∆+V on an appropriate Hilbert space, and their Dirac analogues.

Many problems in Quantum Physics and Chemistry require a precise understanding of the spectra of Schrödinger operators, H = −∆ + V, for various classes of potentials V (x), and in various regimes, especially in the semi-classical and adiabatic ones. The analysis entails determining eigenvalues and eigenvectors and more generally the evolution generated by H, the study of wave operators, and of the “distorted Fourier transform” and its mapping properties. All of these can be interpreted as diagonalization procedures which are especially delicate for non-selfadjoint operators that can arise as linearizations of nonlinear PDEs about solitons.

Indeed, addressing the high-dimensionality of the quantum many-body problems leads to effective, universal approximation schemes given in terms of nonlinear evolution equations in a few dimensions, namely, the Hartree-Fock, Kohn-Sham (density functional theory), Ginzburg-Landau, and Gross-Pitaevskii (nonlinear Schrödinger) equations. Dynamics generated by these equations could be described in terms of interacting spatially localized or periodic solutions (such as solitons, vortices, skyrmions, domain walls, collectively known as coherent structures) plus radiation. Determining the stability of these solutions plays a key role in the understanding of the overall dynamics, formation and stability of crystalline structures. A key step is the linearization of the corresponding equations around the coherent structures which leads to (often, non-selfadjoint) Schrödinger-type operators.

This workshop aims to bring together experts in numerical studies, physical modeling, and applications of theoretical analysis to Schrödinger operators, to compare and contrast the different ideas, various tools and methods developed, as well as their implementation and usage by different communities.