Motivated by applications in quantum chemistry and solid state physics, we apply general results from approximation theory and matrix analysis to the study of the decay properties of spectral projectors associated with large and sparse Hermitian matrices. Our theory leads to a rigorous proof of the exponential off-diagonal decay ("nearsightedness") for the density matrix of gapped systems at zero electronic temperature in both orthogonal and non-orthogonal representations, thus providing a firm theoretical basis for the possibility of linear scaling methods in electronic structure calculations for non-met allic systems. We further discuss the case of density matrices for met allic systems at positive electronic temperature. A few other possible applications are also discussed.
In an earlier paper, W. Kohn had qualitatively introduced the concept of “nearsightedness” of electrons in many-atom systems. It can be viewed as underlying such important ideas as Pauling's “chemical bond,” “transferability,” and Yang's computational principle of “divide and conquer.” It describes the fact that, for fixed chemical potential, local electronic properties, such as the density n(r), depend significantly on the effective external potential only at nearby points. Changes of that potential, no matter how large, beyond a distance R have limited effects on local electronic properties, which rapidly tend to zero as a function of R. In the present paper, the concept is first sharpened for representative models of uncharged fermions moving in external potentials, and then the effects of electron–electron interactions and of perturbing external charges are discussed. A partial answer (Theorem 1 below) to a problem concerning analytic and periodic families of projections in Hilbert spaces is given. As a consequence the existence of exponentially localised Wannier functions corresponding to nondegenerated bands of arbitrary three-dimensional crystals is proved. The concept of nearsightedeness of electronic matter (NEM) was introduced by W. Kohn in 1996 as the physical principal underlining Yang's electronic structure alghoritm of divide and conquer. It describes the fact that, for fixed chemical potential, local electronic properties at a point @math , like the density @math , depend significantly on the external potential @math only at nearby points. Changes @math of that potential, no matter how large , beyond a distance @math , have limited effects on local electronic properties, which tend to zero as function of @math . This remains true even if the changes in the external potential completely surrounds the point @math . NEM can be quantitatively characterized by the nearsightedness range, @math , defined as the smallest distance from @math , beyond which any change of the external potential produces a density change, at @math , smaller than a given @math . The present paper gives a detailed analysis of NEM for periodic met als and insulators in 1D and includes sharp, explicit estimates of the nearsightedness range. Since NEM involves arbitrary changes of the external potential, strong, even qualitative changes can occur in the system, such as the discretization of energy bands or the complete filling of the insulating gap of an insulator with continuum spectrum. In spite of such drastic changes, we show that @math has only a limited effect on the density, which can be quantified in terms of simple parameters of the unperturbed system. This paper communicates recent results in the theory of complex symmetric operators and shows, through two non-trivial examples, their potential usefulness in the study of Schrodinger operators. In particular, we propose a formula for computing the norm of a compact complex symmetric operator. This observation is applied to two concrete problems related to quantum mechanical systems. First, we give sharp estimates on the exponential decay of the resolvent and the single-particle density matrix for Schrodinger operators with spectral gaps. Second, we provide new ways of evaluating the resolvent norm for Schrodinger operators appearing in the complex scaling theory of resonances. The exponential localization of Wannier functions in two or three dimensions is proven for all insulators that display time-reversal symmetry, settling a long-standing conjecture. Our proof relies on the equivalence between the existence of analytic quasi-Bloch functions and the nullity of the Chern numbers (or of the Hall current) for the system under consideration. The same equivalence implies that Chern insulators cannot display exponentially localized Wannier functions. An explicit condition for the reality of the Wannier functions is identified.  The one-dimensional Schr "odinger equation with a periodic and symmetric potential is considered, under the assumption that the energy bands do not intersect. The Bloch waves, @math , and energy bands, @math , are studied as functions of the complex variable, @math . In the complex plane, they are branches of multivalued analytic and periodic functions, @math , and @math , with branch points, @math , off the real axis. A simple procedure is described for locating the branch points. Application is made to the power series and Fourier series developments of these functions. The analyticity and periodicity of @math has some consequences for the form of the Wannier functions. In particular, it is shown that for each band there exists one and only one Wannier function which is real, symmetric or antisymmetric under an appropriate reflection, and falling off exponentially with distance. The rate of falloff is determined by the distance of the branch points @math from the real axis.