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.
Further papers of interest include the work by Prodan, Kohn, and collaborators @cite_6 @cite_79 @cite_118 . From the mathematical standpoint, the most satisfactory results are perhaps those presented in @cite_79 . In this paper, the authors use norm estimates for complex symmetric operators in Hilbert space to obtain sharp exponential decay estimates for the resolvents of rather general Hamiltonians with spectral gap. Using the contour integral representation formula ), these estimates yield (for sufficiently large separations) exponential spatial decay bounds of the form for a broad class of insulators. A lower bound on the decay rate @math (also known as the decay length or inverse correlation length ) is derived, and the behavior of @math as a function of the spectral gap @math is examined.
@cite_79 @cite_118 @cite_6
2963707073 2048738839 2135775236 2055064119
We derive optimal estimates in stochastic homogenization of linear elliptic equations in divergence form in dimensions @math . In previous works we studied the model problem of a discrete elliptic equation on @math . Under the assumption that a spectral gap estimate holds in probability, we proved that there exists a stationary corrector field in dimensions @math and that the energy density of that corrector behaves as if it had finite range of correlation in terms of the variance of spatial averages - the latter decays at the rate of the central limit theorem. In this article we extend these results, and several other estimates, to the case of a continuum linear elliptic equation whose (not necessarily symmetric) coefficient field satisfies a continuum version of the spectral gap estimate. In particular, our results cover the example of Poisson random inclusions. Consider a system governed by the time-dependent Schrodinger equation initialized in its ground state. When subjected to weak (size e) parametric forcing by an “ionizing field” (time-varying), the state decays with advancing time due to coupling of the bound state to radiation modes. The decay rate of this metastable state is governed by Fermi’s golden rule, Γ[V], which depends on the potential V and the details of the forcing. We pose the following potential design problem: find Vopt which minimizes Γ[V] (maximizes the lifetime of the state) over an admissible class of potentials with fixed spatial support. We formulate this problem as a constrained optimization problem and prove that an admissible optimal solution exists. Then, using quasi-Newton methods, we compute locally optimal potentials. These have the structure of a truncated periodic potential with a localized defect. In contrast to optimal structures for other spectral optimization problems, our optimizing potentials appear to be interior point... 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. This paper improves upon best-known guarantees for exact reconstruction of a sparse signal f from a small universal sample of Fourier measurements. The method for reconstruction that has recently gained momentum in the sparse approximation theory is to relax this highly nonconvex problem to a convex problem and then solve it as a linear program. We show that there exists a set of frequencies Ω such that one can exactly reconstruct every r-sparse signal f of length n from its frequencies in Ω, using the convex relaxation, and Ω has size k(r, n) = O(r log(n)·log 2 (r) log(r logn)) = O(r log 4 n ). A random set Ω satisfies this with high probability. This estimate is optimal within the log log n and log 3 r factors. We also give a relatively short argument for a similar problem with k(r, n) ≈ r[12 + 8 log(n r)] Gaussian measurements. We use methods of geometric functional analysis and probability theory in Banach spaces, which makes our arguments quite short.