Sharp Spectral Asymptotics for 2-dimensional Schr\"odinger operator with a strong magnetic field. Note about forgotten generic case
I consider magnetic Schr\"odinger operator in dimension $d=2$ assuming that coefficients are smooth and magnetic field is non-degenerating. Then I extend the remainder estimate $O(\mu^{-1}h^{-1}+1)$ derived in \cite{Ivr1} for the case when $V/F$ has no stationary points to the case when it has non-degenerating stationary points. If some of them are saddles and $\mu^3h\ge 2$ then asymptotics contains correction terms of magnitude $\mu^{-1}h^{-1}|\log \mu^3 h|$.
Sharp spectral asymptotics for generic 4-dimensional Schrödinger operator with the strong magnetic field
I consider 4-dimensional Schrödinger operator with the generic non-degenerating magnetic field and for a generic potential I derive spectral asymptotics with the remainder estimate O(μ−1h−3) and the principal part ≍ h−4 where h ≪ 1 is Planck constant and μ ≫ 1 is the intensity of the magnetic field. For general potentials remainder estimate O(μ−1h−3 + μ2h−2) is achieved.
Imaging the Internal Structure of an Alpine Glacier via L-Band Airborne SAR Tomography
In this paper, we report results from the analysis of 3-D L-band airborne synthetic aperture radar (SAR) acquisitions acquired in March 2014 over the Mittelbergferner glacier, Austrian Alps, during the European Space Agency (ESA) campaign AlpTomoSAR. The campaign included coincident in situ measurements of snow and ice properties and ground-penetrating radar (GPR) data acquired at 600 and 200 MHz over a total length of 18 km. Radar data were acquired by repeatedly flying an L-band SAR along an oval racetrack at an altitude of about 1300 m over the glacier, such that two data stacks from opposite views are obtained. Data from all passes were coherently combined to achieve 3-D resolution capabilities, resulting in the generation of 3-D tomographic SAR (TomoSAR) cubes, where each voxel represents L-band radar reflectivity from a particular location in the 3-D space at a spatial resolution on the order of meters. TomoSAR cubes were finally corrected to account for wave propagation velocity into the ice, which was a necessary step to associate the observed features with their geometrical location, hence enabling a direct comparison to GPR data. The TomoSAR cubes show the complexity of the glacier subsurface scattering. Most areas are characterized by surface scattering in proximity of the ice surface, plus a complex pattern of in-depth volumetric scattering beneath and scattering at the ice/bedrock interface. Various subsurface features observed in GPR transects at 200 MHz clearly showed up in TomoSAR sections as well, particularly firn bodies, crevasses, layer transitions, and bedrock reflection down to 50 m below the ice surface.