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.
A few observations are in order: first, the decay result ) is asymptotic, that is, it implies fast decay at sufficiently large distances @math only. Second, ) is consistent not only with strict exponential decay, but also with decay of the form @math where @math is arbitrary (positive or negative) and @math . Hence, the actual decay could be faster, but also slower, than exponential. Since the result in ) provides only an estimate (rather than an upper bound) for the density matrix in real space, it is not easy to use in actual calculations. To be fair, such practical aspects were not discussed by Kohn until much later (see, e.g., @cite_12 ). Also, later work showed that the asymptotic regime is achieved already for distances of the order of 1-2 lattice constants, and helped clarify the form of the power-law prefactor, as discussed below.
@cite_12
2113726570 2065748556 2046838574 2031810620
Analytic results for the asymptotic decay of the electron density matrix in insulators have been obtained in all three dimensions ( @math ) for a tight-binding model defined on a simple cubic lattice. The anisotropic decay length is shown to be dependent on the energy parameters of the model. The existence of the power-law prefactor, @math , is demonstrated. Analytical results for the asymptotic spatial decay of the density matrix p(r,r') in the tight-binding model of the two-dimensional met al are presented. In various dimensions D, it is found analytically and numerically that the density matrix decays with distance according to the power law ρ(r,r')‖r-r'‖ - ( D + 1 ) 2 . We prove sharp pointwise t−3 decay for scalar linear perturbations of a Schwarzschild black hole without symmetry assumptions on the data. We also consider electromagnetic and gravitational perturbations for which we obtain decay rates t−4, and t−6, respectively. We proceed by decomposition into angular momentum l and summation of the decay estimates on the Regge-Wheeler equation for fixed l. We encounter a dichotomy: the decay law in time is entirely determined by the asymptotic behavior of the Regge-Wheeler potential in the far field, whereas the growth of the constants in l is dictated by the behavior of the Regge-Wheeler potential in a small neighborhood around its maximum. In other words, the tails are controlled by small energies, whereas the number of angular derivatives needed on the data is determined by energies close to the top of the Regge-Wheeler potential. This dichotomy corresponds to the well-known principle that for initial times the decay reflects the presence of complex resonances generated by the potential maximum, whereas for later times the tails are determined by the far field. However, we do not invoke complex resonances at all, but rely instead on semiclassical Sigal-Soffer type propagation estimates based on a Mourre bound near the top energy. Abstract At the heart of this article will be the study of a branching Brownian motion (BBM) with killing , where individual particles move as Brownian motions with drift − ρ , perform dyadic branching at rate β and are killed on hitting the origin. Firstly, by considering properties of the right-most particle and the extinction probability, we will provide a probabilistic proof of the classical result that the ‘one-sided’ FKPP travelling-wave equation of speed − ρ with solutions f : [ 0 , ∞ ) → [ 0 , 1 ] satisfying f ( 0 ) = 1 and f ( ∞ ) = 0 has a unique solution with a particular asymptotic when ρ 2 β , and no solutions otherwise. Our analysis is in the spirit of the standard BBM studies of [S.C. Harris, Travelling-waves for the FKPP equation via probabilistic arguments, Proc. Roy. Soc. Edinburgh Sect. A 129 (3) (1999) 503–517] and [A.E. Kyprianou, Travelling wave solutions to the K-P-P equation: alternatives to Simon Harris' probabilistic analysis, Ann. Inst. H. Poincare Probab. Statist. 40 (1) (2004) 53–72] and includes an intuitive application of a change of measure inducing a spine decomposition that, as a by product, gives the new result that the asymptotic speed of the right-most particle in the killed BBM is 2 β − ρ on the survival set. Secondly, we introduce and discuss the convergence of an additive martingale for the killed BBM, W λ , that appears of fundamental importance as well as facilitating some new results on the almost-sure exponential growth rate of the number of particles of speed λ ∈ ( 0 , 2 β − ρ ) . Finally, we prove a new result for the asymptotic behaviour of the probability of finding the right-most particle with speed λ > 2 β − ρ . This result combined with Chauvin and Rouault's [B. Chauvin, A. Rouault, KPP equation and supercritical branching Brownian motion in the subcritical speed area. Application to spatial trees, Probab. Theory Related Fields 80 (2) (1988) 299–314] arguments for standard BBM readily yields an analogous Yaglom-type conditional limit theorem for the killed BBM and reveals W λ as the limiting Radon–Nikodým derivative when conditioning the right-most particle to travel at speed λ into the distant future.