Higher Order Quantum Superintegrability: a new "Painlev\'e conjecture"

We review recent results on superintegrable quantum systems in a two-dimensional Euclidean space with the following properties. They are integrable because they allow the separation of variables in Cartesian coordinates and hence allow a specific integral of motion that is a second order polynomial in the momenta. Moreover, they are superintegrable because they allow an additional integral of order $N>2$. Two types of such superintegrable potentials exist. The first type consists of "standard potentials" that satisfy linear differential equations. The second type consists of "exotic potentials" that satisfy nonlinear equations. For $N= 3$, 4 and 5 these equations have the Painlev\'e property. We conjecture that this is true for all $N\geq3$. The two integrals X and Y commute with the Hamiltonian, but not with each other. Together they generate a polynomial algebra (for any $N$) of integrals of motion. We show how this algebra can be used to calculate the energy spectrum and the wave functions.Comment: 23 pages, submitted as a contribution to the monographic volume "Integrability, Supersymmetry and Coherent States", a volume in honour of Professor V\'eronique Hussin. arXiv admin note: text overlap with arXiv:1703.0975

A New QCD facility at the M2 beam line of the CERN SPS: COMPASS++/AMBER

In this Letter of Intent, we propose a broad experimental programme for the ``New QCD facility at the M2 beam line of the CERN SPS''. This unrivalled installation will provide the site for a great variety of measurements to address fundamental issues of Quantum Chromodynamics, which are expected to lead to significant improvements in the understanding of QCD as our present theory of strong interactions. The proposed measurements cover the range from lowest-$Q^2$ physics as the determination of the proton radius by elastic muon-proton scattering, over average-$Q^2$-reactions to study hadron spectroscopy, to high-$Q^2$ hadron-structure investigations using the Drell-Yan process and Deeply Virtual Compton Scattering