Variational discrete variable representation for excitons on a lattice

We construct numerical basis function sets on a lattice, whose spatial extension is scalable from single lattice sites to the continuum limit. They allow us to compute small and large bound states with comparable, moderate effort. Adopting concepts of discrete variable representations, a diagonal form of the potential term is achieved through a unitary transformation to Gaussian quadrature points. Thereby the computational effort in three dimensions scales as the fourth instead of the sixth power of the number of basis functions along each axis, such that it is reduced by two orders of magnitude in realistic examples. As an improvement over standard discrete variable representations, our construction preserves the variational principle. It allows for the calculation of binding energies, wave functions, and excitation spectra. We use this technique to study central-cell corrections for excitons beyond the continuum approximation. A discussion of the mass and spectrum of the yellow exciton series in the cuprous oxide, which does not follow the hydrogenic Rydberg series of Mott-Wannier excitons, is given on the basis of a simple lattice model.Comment: 12 pages, 7 figures. Final version as publishe

An a posteriori verification method for generalized real-symmetric eigenvalue problems in large-scale electronic state calculations

An a posteriori verification method is proposed for the generalized real-symmetric eigenvalue problem and is applied to densely clustered eigenvalue problems in large-scale electronic state calculations. The proposed method is realized by a two-stage process in which the approximate solution is computed by existing numerical libraries and is then verified in a moderate computational time. The procedure returns intervals containing one exact eigenvalue in each interval. Test calculations were carried out for organic device materials, and the verification method confirms that all exact eigenvalues are well separated in the obtained intervals. This verification method will be integrated into EigenKernel (https://github.com/eigenkernel/), which is middleware for various parallel solvers for the generalized eigenvalue problem. Such an a posteriori verification method will be important in future computational science.Comment: 15 pages, 7 figure

The spectral density of a difference of spectral projections

Let $H_0$ and $H$ be a pair of self-adjoint operators satisfying some standard assumptions of scattering theory. It is known from previous work that if $\lambda$ belongs to the absolutely continuous spectrum of $H_0$ and $H$, then the difference of spectral projections $$D(\lambda)=1_{(-\infty,0)}(H-\lambda)-1_{(-\infty,0)}(H_0-\lambda)$$ in general is not compact and has non-trivial absolutely continuous spectrum. In this paper we consider the compact approximations $D_\varepsilon(\lambda)$ of $D(\lambda)$, given by $$D_\varepsilon(\lambda)=\psi_\varepsilon(H-\lambda)-\psi_\varepsilon(H_0-\lambda),$$ where $\psi_\varepsilon(x)=\psi(x/\varepsilon)$ and $\psi(x)$ is a smooth real-valued function which tends to $\mp1/2$ as $x\to\pm\infty$. We prove that the eigenvalues of $D_\varepsilon(\lambda)$ concentrate to the absolutely continuous spectrum of $D(\lambda)$ as $\varepsilon\to+0$. We show that the rate of concentration is proportional to $|\log\varepsilon|$ and give an explicit formula for the asymptotic density of these eigenvalues. It turns out that this density is independent of $\psi$. The proof relies on the analysis of Hankel operators.Comment: Final version; to appear in Commun. Math. Physic

Efficient numerical solution of the time fractional diffusion equation by mapping from its Brownian counterpart

The solution of a Caputo time fractional diffusion equation of order $0<\alpha<1$ is expressed in terms of the solution of a corresponding integer order diffusion equation. We demonstrate a linear time mapping between these solutions that allows for accelerated computation of the solution of the fractional order problem. In the context of an $N$-point finite difference time discretisation, the mapping allows for an improvement in time computational complexity from $O\left(N^2\right)$ to $O\left(N^\alpha\right)$, given a precomputation of $O\left(N^{1+\alpha}\ln N\right)$. The mapping is applied successfully to the least-squares fitting of a fractional advection diffusion model for the current in a time-of-flight experiment, resulting in a computational speed up in the range of one to three orders of magnitude for realistic problem sizes.Comment: 9 pages, 5 figures; added references for section