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We prove necessary density conditions for sampling in spectral subspaces of asecond order uniformly elliptic differential operator on $R^d$ with slowlyoscillating symbol. For constant coefficient operators, these are preciselyLandaus necessary density conditions for bandlimited functions, but for moregeneral elliptic differential operators it has been unknown whether such acritical density even exists. Our results prove the existence of a suitablecritical sampling density and compute it in terms of the geometry defined bythe elliptic operator. In dimension 1, functions in a spectral subspace can beinterpreted as functions with variable bandwidth, and we obtain a new criticaldensity for variable bandwidth. The methods are a combination of the spectraltheory and the regularity theory of elliptic partial differential operators,some elements of limit operators, certain compactifications of $R^d $, and thetheory of reproducing kernel Hilbert spaces.

Necessary Density Conditions for Sampling and Interpolation in Spectral Subspaces of Elliptic Differential Operators