Discrete Schrodinger Operators Research Articles

The spectrum of non-local discrete Schrödinger operators with a δ-potential

The behaviour of the spectral edges (embedded eigenvalues and resonances) is discussed at the two ends of the continuous spectrum of non-local discrete Schrodinger operators with a δ-potential. These operators arise by replacing the discrete Laplacian by a strictly increasing C1-function of the discrete Laplacian. The dependence of the results on this function and the lattice dimension are explicitly derived. It is found that while in the case of the discrete Schrodinger operator these behaviours are the same no matter which end of the continuous spectrum is considered, an asymmetry occurs for the non-local cases. A classification with respect to the spectral edge behaviour is also offered.

Effects of discreteness in the Green's function of a hyperbolic medium

We present a theoretical study of the discrete Green's function for the two-dimensional metamaterial, realized as a transmission line array. We demonstrate that the Green's function possesses distinct ripples when the isofrequency contour is of hyperbolic shape. This effect is a characteristic nonlocal feature of the hyperbolic regime and is beyond the effective medium approximation.

Lifshitz asymptotics for percolation Hamiltonians

We study a discrete Laplace operator $\Delta$ on percolation subgraphs of an infinite graph. The ball volume is assumed to grow at most polynomially. We are interested in the behavior of the integrated density of states near the lower spectral edge. If the graph is a Cayley graph we prove that it exhibits Lifshitz tails. If we merely assume that the graph has an exhausting sequence with positive $\delta$-dimensional density, we obtain an upper bound on the integrated density of states of Lifshitz type.

Efficient encoding of texture coordinates guided by mesh geometry

In this paper, we investigate the possibilities of efficient encoding of UV coordinates associated with vertices of a triangle mesh. Since most parametrization schemes attempt to achieve at least some level of conformality, we exploit the similarity of the shapes of triangles in the mesh and in the parametrization. We propose two approaches building on this idea: first, applying a recently proposed generalization of the parallelogram predictor, using the inner angles of mesh triangles corresponding to the UV-space triangles. Second, we propose an encoding method based on discrete Laplace operator, which also allows exploiting the information contained in the mesh geometry to efficiently encode the parametrization. Our experiments show that the proposed approach leads to savings of up to 3 bits per UV vertex, without loss of precision.

Upper semicontinuity of pullback attractors for lattice nonclassical diffusion delay equations under singular perturbations

We consider the following lattice nonclassical diffusion equation with delaysv̇i(t)+λ0vi(t)+(-1)pΔpvi(t)+ε(-1)pΔpv̇i(t)=fi(vi(t-ρ(t)))+gi(t),i∈Z,where ε∈(0,1],λ0 is a positive constant with λ0<1,p is any positive integer and ▵ is the discrete one-dimensional Laplace operator. Under suitable conditions on f and g we prove the existence of pullback attractors for the multi-valued process associated with the ε-small perturbed systems for which the uniqueness of solutions need not hold. Moreover, we compare the dynamics of the original systems and the ε-small perturbed systems, and show that their attractors are “close” in the sense of Hausdorff semidistance.

Application of the discrete Green's function-based antenna simulations for excitation of the total-field/scattered-field interface in the FDTD method

In this article, the discrete Green's function formulation of the finite-difference time-domain (DGF-FDTD) method is proposed for simulation of wire antennas irradiating inhomogeneous dielectric scatterers. Surface equivalence theorem in the discrete domain is used to separate the problem into an inhomogeneous domain and a wire antenna that are simulated with the use of FDTD and DGF-FDTD, respectively. Then, the excitation of the total-field/scattered-field (TFSF) interface within the FDTD domain is computed as a convolution of antenna currents with DGF. Such a radiation is compatible with the grid and can excite the FDTD domain without the erroneous field leakage across the TFSF interface. The developed method is illustrated by canonical problems involving wire antennas that irradiate scatterers. If the DGF length is equal to the number of iterations in a simulation, the developed method perfectly introduces irradiation from wire antennas into the TFSF FDTD domain (assuming infinite numerical precision of computations). © 2014 Wiley Periodicals, Inc. Microwave Opt Technol Lett 56:1949–1953, 2014

A mimetic, semi-implicit, forward-in-time, finite volume shallow water model: comparison of hexagonal–icosahedral and cubed-sphere grids

Abstract. A new algorithm is presented for the solution of the shallow water equations on quasi-uniform spherical grids. It combines a mimetic finite volume spatial discretization with a Crank–Nicolson time discretization of fast waves and an accurate and conservative forward-in-time advection scheme for mass and potential vorticity (PV). The algorithm is implemented and tested on two families of grids: hexagonal–icosahedral Voronoi grids, and modified equiangular cubed-sphere grids. Results of a variety of tests are presented, including convergence of the discrete scalar Laplacian and Coriolis operators, advection, solid body rotation, flow over an isolated mountain, and a barotropically unstable jet. The results confirm a number of desirable properties for which the scheme was designed: exact mass conservation, very good available energy and potential enstrophy conservation, consistent mass, PV and tracer transport, and good preservation of balance including vanishing ∇ × ∇, steady geostrophic modes, and accurate PV advection. The scheme is stable for large wave Courant numbers and advective Courant numbers up to about 1. In the most idealized tests the overall accuracy of the scheme appears to be limited by the accuracy of the Coriolis and other mimetic spatial operators, particularly on the cubed-sphere grid. On the hexagonal grid there is no evidence for damaging effects of computational Rossby modes, despite attempts to force them explicitly.

Singular Density of States Measure for Subshift and Quasi-Periodic Schrödinger Operators

Simon's subshift conjecture states that for every aperiodic minimal subshift of Verblunsky coefficients, the common essential support of the associated measures has zero Lebesgue measure. We disprove this conjecture in this paper, both in the form stated and in the analogous formulation of it for discrete Schr\"odinger operators. In addition we prove a weak version of the conjecture in the Schr\"odinger setting. Namely, under some additional assumptions on the subshift, we show that the density of states measure, a natural measure associated with the operator family and whose topological support is equal to the spectrum, is singular. We also consider one-frequency quasi-periodic Schr\"odinger operators with continuous sampling functions and show that generically, the density of states measure is singular as well.

Positivity of eigenvalues of the two-particle Schrödinger operator on a lattice

We consider the family H(k) of two-particle discrete Schrodinger operators depending on the quasimomentum of a two-particle system k ∈ \(\mathbb{T}^d \), where \(\mathbb{T}^d \) is a d-dimensional torus. This family of operators is associated with the Hamiltonian of a system of two arbitrary particles on the d-dimensional lattice ℤd, d ≥ 3, interacting via a short-range attractive pair potential. We prove that the eigenvalues of the Schrodinger operator H(k) below the essential spectrum are positive for all nonzero values of the quasimomentum k ∈ \(\mathbb{T}^d \) if the operator H(0) is nonnegative. We establish a similar result for the eigenvalues of the Schrodinger operator H+(k), k ∈ \(\mathbb{T}^d \), corresponding to a two-particle system with repulsive interaction.

Multigrid methods for saddle point problems: Stokes and Lamé systems

We develop new multigrid methods for a class of saddle point problems that include the Stokes system in fluid flow and the Lame system in linear elasticity as special cases. The new smoothers in the multigrid methods involve optimal preconditioners for the discrete Laplace operator. We prove uniform convergence of the $$W$$ W -cycle algorithm in the energy norm and present numerical results for $$W$$ W -cycle and $$V$$ V -cycle algorithms.

Analytical Expression for the Time-Domain Discrete Green's Function of a Plane Wave Propagating in the 2-D FDTD Grid

In this letter, a new closed-form expression for the time-domain discrete Green's function (DGF) of a plane wave propagating in the 2-D finite-difference time-domain (FDTD) grid is derived. For the sake of its verification, the time-domain implementation of the analytic field propagator (AFP) technique was developed for the plane wave injection in 2-D total-field/scattered-field (TFSF) FDTD simulations. Such an implementation of AFP requires computations of time-domain DGF for the plane wave with the use of multiple-precision arithmetic. Then, excitations at the TFSF interface can be computed as a time-domain convolution of a source function with DGF. The developed time-domain implementation of AFP demonstrates the leakage error across the TFSF interface around the numerical noise level that validates the correctness of the DGF derivation.

Mathematical Analysis on a Conforming Finite Element Scheme for Advection-Dispersion-Decay Equations on Connected Graphs

Theoretical stability and error analysis on a Conforming Petrov-Galerkin Finite Element (CPGFE) scheme with the fitting technique for solving the Advection-Dispersion-Decay Equations (ADDEs) on connected graphs is performed. This paper is the first research paper that applies the concept of the discrete Green's function (DGF) to error analysis on a numerical scheme for the ADDEs on connected graphs. Firstly, the stability analysis shows that the scheme is unconditionally stable in space for steady problems and is stable in both space and time for unsteady problems if the temporal term is appropriately discretized with a lumping technique. Secondly, basic properties of the DGF on connected graphs, which provide key mathematical tools in the error analysis, are presented. The error analysis with the DGF reveals a direct relationship between the regularity conditions on the known functions and accuracy of the scheme, explicitly indicating that the accuracy of the scheme is strongly influenced by the accuracy of the discretized known functions. The error analysis also shows that the scheme is uniformly-convergent in the L∞-error norm with respect to the diffusivity, which cannot be achieved in the conventional numerical schemes. This unique and remarkable property is a significant advantage of the present CPGFE scheme over the conventional ones.

A Robust Adaptive Grid Method for a System of Two Singularly Perturbed Convection-Diffusion Equations with Weak Coupling

A system of singularly perturbed convection-diffusion equations with weak coupling is considered. The system is first discretized by an upwind finite difference scheme for which an a posteriori error estimate in the maximum norm is constructed. Then the a posteriori error bound is used to design an adaptive gird algorithm. Finally, a first-order rate of convergence, independent of the perturbation parameters, is established by using the theory of the discrete Green's function. Numerical results are presented to illustrate support our theoretical results.

A minimally-resolved immersed boundary model for reaction-diffusion problems

We develop an immersed boundary approach to modeling reaction-diffusion processes in dispersions of reactive spherical particles, from the diffusion-limited to the reaction-limited setting. We represent each reactive particle with a minimally-resolved "blob" using many fewer degrees of freedom per particle than standard discretization approaches. More complicated or more highly resolved particle shapes can be built out of a collection of reactive blobs. We demonstrate numerically that the blob model can provide an accurate representation at low to moderate packing densities of the reactive particles, at a cost not much larger than solving a Poisson equation in the same domain. Unlike multipole expansion methods, our method does not require analytically computed Green's functions, but rather, computes regularized discrete Green's functions on the fly by using a standard grid-based discretization of the Poisson equation. This allows for great flexibility in implementing different boundary conditions, coupling to fluid flow or thermal transport, and the inclusion of other effects such as temporal evolution and even nonlinearities. We develop multigrid-based preconditioners for solving the linear systems that arise when using implicit temporal discretizations or studying steady states. In the diffusion-limited case the resulting linear system is a saddle-point problem, the efficient solution of which remains a challenge for suspensions of many particles. We validate our method by comparing to published results on reaction-diffusion in ordered and disordered suspensions of reactive spheres.

Electron scattering by a crystal layer

We consider the one-particle discrete Schrodinger operator H with a periodic potential perturbed by a function ɛW that is periodic in two variables and exponentially decreasing in the third variable. Here, ɛ is a small parameter. We study the scattering problem for H near the point of extremum with respect to the third quasimomentum coordinate for a certain eigenvalue of the Schrodinger operator with a periodic potential in the cell, in other words, for the small perpendicular component of the angle of particle incidence on the potential barrier ɛW. We obtain simple formulas for the transmission and reflection probabilities.

A discrete Gauss-Green identity for unbounded Laplace operators, and the transience of random walks

On a finite network (connected weighted undirected graph), the relationship between the natural Dirichlet form E and the discrete Laplace operator Δ is given by \(\varepsilon (u,v) = {\langle u,\Delta v\rangle _{{\ell ^2}}}\), where the latter is the usual ℓ 2 inner product. This formula is not generally true for infinite networks; earlier authors have given various conditions under which this formula remains valid. Instead, we extend this formula to arbitrary infinite networks (including the case when Δ is unbounded) by including a new (boundary) term, in parallel with the classical Gauss-Green identity. This tool allows for detailed study of the boundary of the network. We construct a reproducing kernel for the space of functions of finite energy which allows us to specify a dense domain for Δ and give several criteria for the transience of the random walk on the network. The extended Gauss-Green identity and the reproducing kernel also yield a boundary integral representation for harmonic functions of finite energy. The boundary representation is developed further in [24].

Discrete Green's Function Approach for The Analysis of A Dual Band‐Notched Uwb Antenna

ABSTRACTA novel UWB monopole antenna with trident feeding and dual band‐notched characteristics is proposed. The antenna consists of two bottom corner notches for improvement of impedance matching and two U‐shaped slots for rejection of WLAN and WiMAX bands. Discrete Green's function approach has been utilized to numerically achieve time‐domain current distribution on the antenna. A good agreement between simulation and experimental results is observed. © 2013 Wiley Periodicals, Inc. Microwave Opt Technol Lett 55:2168–2174, 2013

Vertical-probe-induced asymmetric dust oscillation in complex plasma

A complex plasma vertical oscillation experiment which modifies the bulk is presented. Spherical, micron-sized particles within a Coulomb crystal levitated in the sheath above the powered lower electrode in a GEC reference cell are perturbed using a probe attached to a Zyvex S100 Nanomanipulator. By oscillating the probe potential sinusoidally, particle motion is found to be asymmetric, exhibiting superharmonic response in one case. Using a simple electric field model for the plasma sheath, including a nonzero electric field at the sheath edge, dust particle charges are found by employing a balance of relevant forces and emission analysis. Adjusting the parameters of the electric field model allowed the change predicted in the levitation height to be compared with experiment. A discrete oscillator Green's function is applied using the derived force, which accurately predicts the particle's motion and allows the determination of the electric field at the sheath edge.

Multi-Region Finite-Difference Time-Domain (MR-FDTD) Based on Domain-Optimal Green's Functions

A multi-region (MR-) finite-difference time-domain (FDTD) scheme for solving two-dimensional sparsely-populated problems based on domain-optimal Green's functions is proposed. The scheme uses a discrete Green's function (DGF) on the FDTD lattice to truncate the local sub-regions and thus reduces reflection error on the local boundary. A continuous Green's function (CGF) is implemented to pass the influence of external fields into each FDTD region which mitigates the numerical dispersion and anisotropy of standard FDTD. Numerical results will demonstrate the accuracy of this approach.

EXPLICIT EXPRESSION OF THE KRAWTCHOUK POLYNOMIAL VIA A DISCRETE GREEN'S FUNCTION

A Krawtchouk polynomial is introduced as the classical Mac-Williams identity, which can be expressed in weight-enumerator-free form of a linear code and its dual code over a Hamming scheme. In this paper we find a new explicit expression for the <TEX>$p$</TEX>-number and the <TEX>$q$</TEX>-number, which are more generalized notions of the Krawtchouk polynomial in the P-polynomial schemes by using an extended version of a discrete Green's function. As corollaries, we obtain a new expression of the Krawtchouk polynomial over the Hamming scheme and the Eberlein polynomial over the Johnson scheme. Furthermore, we find another version of the MacWilliams identity over a Hamming scheme.