On the convergence of eigenfunctions to threshold energy states

We prove the convergence in certain weighted spaces in momentum space of eigenfunctions of H = T-lambda*V as the energy goes to an energy threshold. We do this for three choices of kinetic energy T, namely the non-relativistic Schr"odinger operator, the pseudorelativistc operator sqrt{-\Delta+m^2}-m, and the Dirac operator.Comment: 15 pages; references and comments added (e.g., Remark 3

The electron densities of pseudorelativistic eigenfunctions are smooth away from the nuclei

We consider a pseudorelativistic model of atoms and molecules, where the kinetic energy of the electrons is given by $\sqrt{p^2+m^2}-m$. In this model the eigenfunctions are generally not even bounded, however, we prove that the corresponding one-electron densities are smooth away from the nuclei.Comment: 16 page

Electron Wavefunctions and Densities for Atoms

With a special `Ansatz' we analyse the regularity properties of atomic electron wavefunctions and electron densities. In particular we prove an a priori estimate, $\sup_{y\in B(x,R)}|\nabla\psi(y)| \leq C(R) \sup_{y\in B(x,2R)}|\psi(y)|$ and obtain for the spherically averaged electron density, $\widetilde\rho(r)$, that $\widetilde\rho''(0)$ exists and is non-negative

Hartree-Fock theory for pseudorelativistic atoms

We study the Hartree-Fock model for pseudorelativistic atoms, that is, atoms where the kinetic energy of the electrons is given by the pseudorelativistic operator \sqrt{(pc)^2+(mc^2)^2}-mc^2. We prove the existence of a Hartree-Fock minimizer, and prove regularity away from the nucleus and pointwise exponential decay of the corresponding orbitals

Alone in the sustainable wilderness; transforming sustainable competences and didactics in a design for change education

According to UNESCO (2012) pedagogies associated with Educations for Sustainable Development (ESD) should spur and inspire students to think critically, ask questions and reflect. The assumption is, that pedagogies are moving towards student-centred participatory learning. Still, the educator is at the core of the transition towards developing ESD’s, so changes of the educators’ worldviews and practices are emerging. The educators’ competencies and knowledge on ESD-development becomes central as the question of the what and the how the students are taught becomes more pressing.  Today many sustainable educations still have a high focus on systemic issues (external systems); politics, technology, or socioeconomic structures (Parodi & Tamm, 2018, Wamsler, 2019) and lately the UNESCO (2021) has stressed the need for adapting cognitive, transformative, personal, emotional, dimensions of learning into ESD In a transformative learning setting the educators should provide real-heartfelt experiences generating the students with capacities to reflect critically on both systems and personal design practice methods and help them aligning their methods with their personal emotional values. In doing so, the educators feel left “alone in the Wilderness” and research in the personal dimension of sustainable transformation and connection of this to ESD’s is scarce (Parodi & Tamm, 2018, Wamsler, 2019). On this backdrop, this article provides a reflexive case study of a B.A. level course on “Design for Change” performed from 2019 – 2021 using transformative learning practices and the connected interventions in the form of a reflection tool, the Decoding Creativity Tool (DCT). The data was collected to discover if and how the students could enhance personal sustainable competencies using transformative learning focusing on the personal emotional and creative development and awareness, reflection tools and “visiting”-methods. Implementing transformative learning and ESD’s into educational practices requires radical revisions of the design education system, managerial strategic commitment and involves many levels of the HE’s. It requires both internal and external collaborations for the design educations and could involve developing new didactics and methods where ideas can grow. (Barth & Rieckmann, 2012)

Non-isotropic cusp conditions and regularity of the electron density of molecules at the nuclei

We investigate regularity properties of molecular one-electron densities rho near the nuclei. In particular we derive a representation rho(x)=mu(x)*(e^F(x)) with an explicit function F, only depending on the nuclear charges and the positions of the nuclei, such that mu belongs to C^{1,1}(R^3), i.e., mu has locally essentially bounded second derivatives. An example constructed using Hydrogenic eigenfunctions shows that this regularity result is sharp. For atomic eigenfunctions which are either even or odd with respect to inversion in the origin, we prove that mu is even C^{2,\alpha}(R^3) for all alpha in (0,1). Placing one nucleus at the origin we study rho in polar coordinates x=r*omega and investigate rho'(r,omega) and rho''(r,omega) for fixed omega as r tends to zero. We prove non-isotropic cusp conditions of first and second order, which generalize Kato's classical result.Comment: 19 page

Positivity and lower bounds to the decay of the atomic one-electron density

We investigate properties of the spherically averaged atomic one-electron density rho~(r). For a rho~ which stems from a physical ground state we prove that rho~ > 0. We also give exponentially decreasing lower bounds to rho~ in the case when the eigenvalue is below the corresponding essential spectrum.Comment: 20 page