We prove necessary density conditions for sampling in spectral subspaces of a second order uniformly elliptic differential operator on $R^d$ with slowly oscillating symbol. For constant coefficient operators, these are precisely Landaus necessary density conditions for bandlimited functions, but for more general elliptic differential operators it has been unknown whether such a critical density even exists. Our results prove the existence of a suitable critical sampling density and compute it in terms of the geometry defined by the elliptic operator. In dimension 1, functions in a spectral subspace can be interpreted as functions with variable bandwidth, and we obtain a new critical density for variable bandwidth. The methods are a combination of the spectral theory and the regularity theory of elliptic partial differential operators, some elements of limit operators, certain compactifications of $R^d $, and the theory of reproducing kernel Hilbert spaces.

翻译：在第二顺序的光谱亚空间取样时,我们证明必须具备必要的密度条件,即以低振动符号以美元平均椭圆差操作员以美元为单位对第二顺序的光谱亚空间取样。对于恒定系数操作员来说,这些恰恰是带宽功能限制的Landaus必要密度条件,但对于更普通的椭圆差操作员来说,还不清楚这种临界密度是否甚至存在。我们的结果证明存在一个合适的临界取样密度,并根据椭圆操作员界定的几何测量方法进行计算。在尺寸1中,光谱子空间的功能可以被解释为带宽功能,我们为可变带宽获得一种新的临界密度。这些方法结合了光谱理论和椭圆部分差操作员的常规理论、某些限制操作员的元素、美元的某些紧凑化和再生产内核的Hilbert空间的理论。