Critical Lieb-Thirring bounds in gaps and the generalized Nevai conjecture for finite gap Jacobi matrices

We prove bounds of the form ∑_(e∈I⋂σ_d(H)) dist(e, σ_e(H)^(1/2) ≤ L^1 -norm of a perturbation, where I is a gap. Included are gaps in continuum one-dimensional periodic Schrödinger operators and finite gap Jacobi matrices, where we get a generalized Nevai conjecture about an L^(1)-condition implying a Szegő condition. One key is a general new form of the Birman-Schwinger bound in gaps

Condensation of fermion pairs in a domain

We consider a gas of fermions at zero temperature and low density, interacting via a microscopic two body potential which admits a bound state. The particles are confined to a domain with Dirichlet (i.e. zero) boundary conditions. Starting from the microscopic BCS theory, we derive an effective macroscopic Gross-Pitaevskii (GP) theory describing the condensate of fermion pairs. The GP theory also has Dirichlet boundary conditions. Along the way, we prove that the GP energy, defined with Dirichlet boundary conditions on a bounded Lipschitz domain, is continuous under interior and exterior approximations of that domain.Comment: 43 pages, 1 figur

Future Security Approaches and Biometrics

Threats to information security are proliferating rapidly, placing demanding requirements on protecting tangible and intangible business and individual assets. Biometrics can improve security by replacing or complementing traditional security technologies. This tutorial discusses the strengths and weaknesses of biometrics and traditional security approaches, current and future applications of biometrics, performance evaluation measures of biometric systems, and privacy issues surrounding the new technology

Proof of the Strong Scott Conjecture for Chandrasekhar Atoms

We consider a large neutral atom of atomic number Z, taking relativistic effects into account by assuming the dispersion relation √c²p²+c⁴. We study the behavior of the one-particle ground state density on the length scale Z⁻¹ in the limit Z,c→∞ keeping Z/c fixed and find that the spherically averaged density as well as all individual angular momentum densities separately converge to the relativistic hydrogenic ones. This proves the generalization of the strong Scott conjecture for relativistic atoms and shows, in particular, that relativistic effects occur close to the nucleus. Along the way we prove upper bounds on the relativistic hydrogenic density

The mindful path to compassion in an adult mental health group

A naturalistic study was undertaken within an NHS setting to explore the effectiveness and satisfaction with a Mindfulness-Based Cognitive Therapy and Mindful Self-Compassion group programme in an adult mental health population. Outcome measures and qualitative feedback suggested beneficial effects and high levels of satisfaction

Eigenvalue Bounds for Perturbations of Schrödinger Operators and Jacobi Matrices With Regular Ground States

We prove general comparison theorems for eigenvalues of perturbed Schrödinger operators that allow proof of Lieb–Thirring bounds for suitable non-free Schrödinger operators and Jacobi matrices