Equivalence of Sobolev norms involving generalized Hardy operators

We consider the fractional Schr\"odinger operator with Hardy potential and critical or subcritical coupling constant. This operator generates a natural scale of homogeneous Sobolev spaces which we compare with the ordinary homogeneous Sobolev spaces. As a byproduct, we obtain generalized and reversed Hardy inequalities for this operator. Our results extend those obtained recently for ordinary (non-fractional) Schr\"odinger operators and have an important application in the treatment of large relativistic atoms.Comment: 16 pages; v2 contains improved results for positive coupling constant

Growth, the Environment and Keynes: Reflections on Two Heterodox Schools of Thought

This paper explores the approach of Post Keynesian Economics (PKE) in comparison with ecological economics. While PKE, like all macroeconomics, has failed to address environmental problems it does have many aspects which make compatibility with ecological economics seem feasible. Ecological economics has no specific macroeconomic approach although it has strong implications for economic growth and how this should be controlled, directed and in materials terms limited. We highlight growth as the key area of difference and reflect upon how Keynes himself saw capital accumulation as a means to an end not an end in itself, regarded it as a temporary measure and also was well aware of some of its psychological and social drawbacks.environment, Keynes, post keynesian, ecological economics

The Energy of Heavy Atoms According to Brown and Ravenhall: The Scott Correction

We consider relativistic many-particle operators which - according to Brown and Ravenhall - describe the electronic states of heavy atoms. Their ground state energy is investigated in the limit of large nuclear charge and velocity of light. We show that the leading quasi-classical behavior given by the Thomas-Fermi theory is raised by a subleading correction, the Scott correction. Our result is valid for the maximal range of coupling constants, including the critical one. As a technical tool, a Sobolev-Gagliardo-Nirenberg-type inequality is established for the critical atomic Brown-Ravenhall operator. Moreover, we prove sharp upper and lower bound on the eigenvalues of the hydrogenic Brown-Ravenhall operator up to and including the critical coupling constant.Comment: 42 page

Generalized minimizers of convex integral functionals, Bregman distance, Pythagorean identities

Integral functionals based on convex normal integrands are minimized subject to finitely many moment constraints. The integrands are finite on the positive and infinite on the negative numbers, strictly convex but not necessarily differentiable. The minimization is viewed as a primal problem and studied together with a dual one in the framework of convex duality. The effective domain of the value function is described by a conic core, a modification of the earlier concept of convex core. Minimizers and generalized minimizers are explicitly constructed from solutions of modified dual problems, not assuming the primal constraint qualification. A generalized Pythagorean identity is presented using Bregman distance and a correction term for lack of essential smoothness in integrands. Results are applied to minimization of Bregman distances. Existence of a generalized dual solution is established whenever the dual value is finite, assuming the dual constraint qualification. Examples of `irregular' situations are included, pointing to the limitations of generality of certain key results

Mueller's Exchange-Correlation Energy in Density-Matrix-Functional Theory

The increasing interest in the Mueller density-matrix-functional theory has led us to a systematic mathematical investigation of its properties. This functional is similar to the Hartree-Fock functional, but with a modified exchange term in which the square of the density matrix \gamma(X, X') is replaced by the square of \gamma^{1/2}(X, X'). After an extensive introductory discussion of density-matrix-functional theory we show, among other things, that this functional is convex (unlike the HF functional) and that energy minimizing \gamma's have unique densities \rho(x), which is a physically desirable property often absent in HF theory. We show that minimizers exist if N \leq Z, and derive various properties of the minimal energy and the corresponding minimizers. We also give a precise statement about the equation for the orbitals of \gamma, which is more complex than for HF theory. We state some open mathematical questions about the theory together with conjectured solutions.Comment: Latex, 42 pages, 1 figure. Minor error in the proof of Prop. 2 correcte