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We analyze spectral properties of the operator <i>H</i> = <i></i>²/<i>x</i>² <i></i>²/<i>y</i>² + <i></i>²<i>y</i>² <i>y</i>²<i>V</i>(<i>xy</i>) in <i>L</i>²(²), where <i></i> 0 and <i>V </i> 0 is a compactly supported and sufficiently regular potential. It is known that the spectrum of <i>H</i> depends on the one-dimensional Schrdinger operator <i>L</i> = <i>d</i>²/<i>dx</i>² + <i></i>² <i>V</i>(<i>x</i>) and it changes substantially as inf(<i>L</i>) switches sign. We prove that in the critical case, inf(<i>L</i>) = 0, the spectrum of <i>H</i> is purely essential and covers the interval [0, ). In the subcritical case, inf <i></i>(<i>L</i>) &gt; 0, the essential spectrum starts from <i></i> and there is a nonvoid discrete spectrum in the interval [0, <i></i>). We also derive a bound on the corresponding eigenvalue moments.

Different from conventional spaceborne or airborne synthetic aperture radar (SAR) with optimal aperture length, an imaging radar with highly suboptimal aperture length acquires the data in short bursts by a geometry spreading over a large range. A polarlike or pseudopolar format grid is introduced to sample data close to the resolution, which presents the design of a separable kernel for efficient FFT implementation. The proposed imaging algorithm formulates the reflectivity image of the target scene as an interpolation-free double image series expansion with two usual approximation-induced phase error terms being taken into account, whereby more generalized application scenarios with high frequency, large bandwidth or larger aperture length for imaging a target scene located within either the far-field or the near-field of the radar aperture are processable with high accuracy. In addition, convergence

In this paper, we study the global well-posedness of the 2D compressible NavierStokes equations with large initial data and vacuum. It is proved that if the shear viscosity<i></i> is a positive constant and the bulk viscosity <i></i> is the power function of the density, that is, <i></i>(<i></i>) = <i></i> ^<i></i> with <i></i> &gt; 3, then the 2D compressible NavierStokes equations with the periodic boundary conditions on the torus T 2 admit a unique global classical solution (<i>, u</i>) which may contain vacuums in an open set of T 2 . Note that the initial data can be arbitrarily large to contain vacuum states.