Complete asymptotic expansions for eigenvalues of Dirichlet Laplacian in thin three-dimensional rods

We consider Dirichlet Laplacian in a thin curved three-dimensional rod. The rod is finite. Its cross-section is constant and small, and rotates along the reference curve in an arbitrary way. We find a two-parametric set of the eigenvalues of such operator and construct their complete asymptotic expansions. We show that this two-parametric set contains any prescribed number of the first eigenvalues of the considered operator. We obtain the complete asymptotic expansions for the eigenfunctions associated with these first eigenvalues

Neutrino dispersion in external magnetic fields

We calculate the neutrino self-energy operator Sigma (p) in the presence of a magnetic field B. In particular, we consider the weak-field limit e B << m_\ell^2, where m_\ell is the charged-lepton mass corresponding to the neutrino flavor \nu_\ell, and we consider a "moderate field" m_\ell^2 << e B << m_W^2. Our results differ substantially from the previous literature. For a moderate field, we show that it is crucial to include the contributions from all Landau levels of the intermediate charged lepton, not just the ground-state. For the conditions of the early universe where the background medium consists of a charge-symmetric plasma, the pure B-field contribution to the neutrino dispersion relation is proportional to (e B)^2 and thus comparable to the contribution of the magnetized plasma.Comment: 9 pages, 1 figure, revtex. Version to appear in Phys. Rev. D (presentation improved, reference list revised, numerical error in Eq.(41) corrected, conclusions unchanged

Refraction of fast Ne atoms in the attractive well of LiF(001) surface

Ne atoms with energies up to 3 keV are diffracted under grazing angles of incidence from a LiF(001) surface. For a small momentum component of the incident beam perpendicular to the surface, we observe an increase of the elastic rainbow angle together with a broadening of the inelastic scattering profile. We interpret these two effects as the refraction of the atomic wave in the attractive part of the surface potential. We use a fast, rigorous dynamical diffraction calculation to find a projectile-surface potential model that enables a quantitative reproduction of the experimental data for up to ten diffraction orders. This allows us to extract an attractive potential well depth of 10.4 meV. Our results set a benchmark for more refined surface potential models which include the weak Van der Waals region, a long-standing challenge in the study of atom-surface interactions

Homogenization of the planar waveguide with frequently alternating boundary conditions

We consider Laplacian in a planar strip with Dirichlet boundary condition on the upper boundary and with frequent alternation boundary condition on the lower boundary. The alternation is introduced by the periodic partition of the boundary into small segments on which Dirichlet and Neumann conditions are imposed in turns. We show that under the certain condition the homogenized operator is the Dirichlet Laplacian and prove the uniform resolvent convergence. The spectrum of the perturbed operator consists of its essential part only and has a band structure. We construct the leading terms of the asymptotic expansions for the first band functions. We also construct the complete asymptotic expansion for the bottom of the spectrum