Two-body quantum mechanical problem on spheres

The quantum mechanical two-body problem with a central interaction on the sphere ${\bf S}^{n}$ is considered. Using recent results in representation theory an ordinary differential equation for some energy levels is found. For several interactive potentials these energy levels are calculated in explicit form.Comment: 41 pages, no figures, typos corrected; appendix D was adde

CXCR2 deficient mice display macrophage-dependent exaggerated acute inflammatory responses

CXCR2 is an essential regulator of neutrophil recruitment to inflamed and damaged sites and plays prominent roles in inflammatory pathologies and cancer. It has therefore been highlighted as an important therapeutic target. However the success of the therapeutic targeting of CXCR2 is threatened by our relative lack of knowledge of its precise in vivo mode of action. Here we demonstrate that CXCR2-deficient mice display a counterintuitive transient exaggerated inflammatory response to cutaneous and peritoneal inflammatory stimuli. In both situations, this is associated with reduced expression of cytokines associated with the resolution of the inflammatory response and an increase in macrophage accumulation at inflamed sites. Analysis using neutrophil depletion strategies indicates that this is a consequence of impaired recruitment of a non-neutrophilic CXCR2 positive leukocyte population. We suggest that these cells may be myeloid derived suppressor cells. Our data therefore reveal novel and previously unanticipated roles for CXCR2 in the orchestration of the inflammatory response

Generalized Ornstein–Uhlenbeck semigroups in weighted Lp-spaces on Riemannian manifolds

Let E be a Hermitian vector bundle over a Riemannian manifold M with metric g, and let ∇ be a metric covariant derivative on E. We study the generalized Ornstein–Uhlenbeck differential expression P∇=∇†∇u+∇(dϕ)♯u−∇Xu+Vu, where ∇† is the formal adjoint of ∇, (dϕ)♯ is the vector field corresponding to dϕ via g, X is a smooth real vector field on M, and V is a self-adjoint locally integrable section of the endomorphsim bundle EndE. In the setting of a geodesically complete M, we establish a sufficient condition for the equality of the maximal and minimal realizations of P∇ in the (weighted) space ΓLμp(E) of Lμp-type sections of E, where 1<p<∞ and dμ=e−ϕdνg, with νg being the usual volume measure. Furthermore, we show that (the negative of) the maximal realization −Hp,max generates an analytic quasi-contractive semigroup in ΓLμp(E), 1<p<∞. Additionally, in the same context, we establish a coercivity estimate, which leads to the so-called separation property of the covariant Schrödinger operator (that is, P∇ with X≡0 and ϕ≡0) in the (unweighted) space ΓLp(E), 1<p<∞. With the generation result at our disposal, we describe a Feynman–Kac representation for the Lμp-semigroup generated by Hp,max, 1<p<∞. For the Ornstein–Uhlenbeck differential expression acting on functions, that is, Pd=Δu+(dϕ)♯u−Xu+Vu, where Δ is the (non-negative) scalar Laplacian on M and V is a locally integrable real-valued function, we consider another way of realizing Pd as an operator in Lμp(M) and, by imposing certain geometric conditions on M, we prove another semigroup generation result. The study of the mentioned realization of Pd depends, among other things, on the fulfillment of the so-called Lp-Calderón–Zygmund inequality on M.Ognjen Milatovic, Hemanth Saratchandra

Self-adjointness of the Gaffney Laplacian on Vector Bundles

We study the Gaffney Laplacian on a vector bundle equipped with a compatible metric and connection over a Riemannian manifold that is possibly geodesically incomplete. Under the hypothesis that the Cauchy boundary is polar, we demonstrate the self-adjointness of this Laplacian. Furthermore, we show that negligible boundary is a necessary and sufficient condition for the self-adjointness of this operator

Self-Adjoint Extensions of Discrete Magnetic Schrödinger Operators

Using the concept of an intrinsic metric on a locally finite weighted graph, we give sufficient conditions for the magnetic Schrödinger operator to be essentially self-adjoint. The present paper is an extension of some recent results proven in the context of graphs of bounded degree. © 2013 Springer Basel