  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.
