Search results for "Lanczos algorithm"
showing 5 items of 5 documents
Computation of a few smallest eigenvalues of elliptic operators using fast elliptic solvers
2001
The computation of a few smallest eigenvalues of generalized algebraic eigenvalue problems is studied. The considered problems are obtained by discretizing self-adjoint second-order elliptic partial differential eigenvalue problems in two- or three-dimensional domains. The standard Lanczos algorithm with the complete orthogonalization is used to compute some eigenvalues of the inverted eigenvalue problem. Under suitable assumptions, the number of Lanczos iterations is shown to be independent of the problem size. The arising linear problems are solved using some standard fast elliptic solver. Numerical experiments demonstrate that the inverted problem is much easier to solve with the Lanczos…
SPATIAL MULTIFRACTALITY OF ELECTRONIC STATES AND THE METAL-INSULATOR TRANSITION IN DISORDERED SYSTEMS
1993
For the investigation of the spatial behavior of electronic wave functions in disordered systems, we employ the Anderson model of localization. The eigenstates of the corresponding Hamiltonian are calculated numerically by means of the Lanczos algorithm and are analyzed with respect to their spatial multifractal properties. We find that the wave functions show spatial multifractality for all parameter cases not too far away from the metal-insulator transition (MIT) which separates localized from extended states in this model. Exactly at the MIT, multifractality is expected to exist on all length scales larger than the lattice spacing. It is found that the corresponding singularity spectrum…
Multifractal wave functions at the Anderson transition.
1991
Electronic wave functions in disordered systems are studied within the Anderson model of localization. At the critical disorder in 3D we diagonalize very large (103 823\ifmmode\times\else\texttimes\fi{}103 823) secular matrices by means of the Lanczos algorithm. On all length scales the obtained strong spatial fluctuations of the amplitude of the eigenstates display a multifractal character, reflected in the set of generalized fractal dimensions and the singularity spectrum of the fractal measure. An analysis of 1D systems shows multifractality too, in contrast to previous claims.
Benchmark calculations of electromagnetic sum rules with a symmetry-adapted basis and hyperspherical harmonics
2020
We demonstrate the ability to calculate electromagnetic sum rules with the \textit{ab initio} symmetry-adapted no-core shell model. By implementing the Lanczos algorithm, we compute non-energy weighted, energy weighted, and inverse energy weighted sum rules for electric monopole, dipole, and quadrupole transitions in $^4$He using realistic interactions. We benchmark the results with the hyperspherical harmonics method and show agreement within $2\sigma$, where the uncertainties are estimated from the use of the many-body technique. We investigate the dependence of the results on three different interactions, including chiral potentials, and we report on the $^4$He electric dipole polarizabi…
Determination of the mobility edge in the Anderson model of localization in three dimensions by multifractal analysis.
1995
We study the Anderson model of localization in three dimensions with different probability distributions for the site energies. Using the Lanczos algorithm we calculate eigenvectors for different model parameters like disorder and energy. From these we derive the singularity spectrum typically used for the characterization of multifractal objects. We demonstrate that the singularity spectrum at the critical disorder, which determines the mobility edge at the band center, is independent of the employed probability distribution. Assuming that this singularity spectrum is universal for the metal-insulator transition regardless of specific parameters of the model we establish a straightforward …