Discrete Schrodinger Operators Research Articles

Inverse Scattering Theory for Discrete Schrödinger Operators on the Hexagonal Lattice

We consider the spectral theory and inverse scattering problem for discrete Schrodinger operators on the hexagonal lattice. We give a procedure for reconstructing finitely supported potentials from the scattering matrices for all energies. The same procedure is applicable for the inverse scattering problem on the triangle lattice.

On the construction and use of linear low-dimensional ventilation models

The construction of fast reliable low-dimensional models is important for monitoring and control of ventilation applications. We employ a discrete Green's function approach to derive a linear low-dimensional ventilation model directly from the governing equations for indoor ventilation (i.e., the Navier-Stokes equations supplemented with a transport equation for indoor-pollutant concentration). It is shown that the flow equations decouple from the concentration equation when the ratio α of air-mass-flow rate to pollutant-mass-flow rate increases to infinity. A low-dimensional discrete representation of the Green's function of the concentration equation can then be constructed, based on either numerical simulations or experiments. This serves as a linear model that allows for the reconstruction of concentration fields resulting from any type of pollutant-source distribution. We employ a suite of Reynolds-averaged Navier-Stokes (RANS) simulations to illustrate the methodology. We focus on a simple benchmark ventilation case under constant-density conditions. Discrete linear ventilation models for the concentration are then derived and compared with coupled RANS simulations. An analysis of errors in the discrete linear model is presented: dependence of the error on the (low-dimensional) resolution in the discrete model is quantified, and errors introduced by too low values of α are also investigated. The paper introduces the derivation and construction of linear low-dimensional ventilation models, which allow reconstructing concentration fields resulting from any type of indoor-pollutant-source distribution. Once constructed, these ventilation models are very efficient to estimate indoor contaminant concentration distributions, compared to direct CFD simulation approaches. Therefore, these models can facilitate monitoring and control of ventilation systems, to remove indoor contaminants.

Existence and analyticity of bound states of a two-particle Schrödinger operator on a lattice

We consider the two-particle discrete Schrodinger operator Hμ(K) corresponding to a system of two arbitrary particles on a d-dimensional lattice ℤd, d ≥ 3, interacting via a pair contact repulsive potential with a coupling constant μ > 0 (\(K \in \mathbb{T}^d\) is the quasimomentum of two particles). We find that the upper (right) edge of the essential spectrum can be either a virtual level (for d = 3, 4) or an eigenvalue (for d ≥ 5) of Hμ (K). We show that there exists a unique eigenvalue located to the right of the essential spectrum, depending on the coupling constant μ and the two-particle quasimomentum K. We prove the analyticity of the corresponding eigenstate and the analyticity of the eigenvalue and the eigenstate as functions of the quasimomentum \(K \in \mathbb{T}^d\) in the domain of their existence.

Lattice-Boltzmann and meshless point collocation solvers for fluid flow and conjugate heat transfer

SUMMARY The applicability and performance of the lattice-Boltzmann (LB) and meshless point collocation methods as CFD solvers in flow and conjugate heat transfer processes are investigated in this work. Lid-driven cavity flow and flow in a slit with an obstacle including heat transfer are considered as case studies. A comparison of the computational efficiency accuracy of the two methods with that of a finite volume method as implemented in a commercial package (ANSYS CFX, ANSYS Inc., Canonsburg, PA) is made. Utilizing the analogy between heat and mass transfer, an advection–diffusion LB model was adopted to simulate the heat transfer part of the slit flow problem followed by a rigorous mapping of the mass transfer variables to the heat transfer quantities of interest, thus circumventing the need for a thermal LB model. Direct comparison among the results of the three methods revealed excellent agreement over a wide range of Reynolds and Prandtl number values. Furthermore, an integrated computational scheme is proposed, utilizing the rapid convergence of the LB model in the flow part of the conjugate heat transfer problem with that of the meshless collocation method for the heat transfer part. The meshless treatment remains sufficiently rapid even for conduction-controlled processes in contrast to the LB method, which is very rapid in the convectioncontrolled case only. A single, common computational grid, composed of regularly distributed nodes is used, saving significant computational and coding time and ensuring convergence of the discrete Laplacian operator in the heat transfer part of the computations. Copyright © 2012 John Wiley & Sons, Ltd.

Modal shape analysis beyond Laplacian

In recent years, substantial progress in shape analysis has been achieved through methods that use the spectra and eigenfunctions of discrete Laplace operators. In this work, we study spectra and eigenfunctions of discrete differential operators that can serve as an alternative to the discrete Laplacians for applications in shape analysis. We construct such operators as the Hessians of surface energies, which operate on a function space on the surface, or of deformation energies, which operate on a shape space. In particular, we design a quadratic energy such that, on the one hand, its Hessian equals the Laplace operator if the surface is a part of the Euclidean plane, and, on the other hand, the Hessian eigenfunctions are sensitive to the extrinsic curvature (e.g. sharp bends) on curved surfaces. Furthermore, we consider eigenvibrations induced by deformation energies, and we derive a closed form representation for the Hessian (at the rest state of the energy) for a general class of deformation energies. Based on these spectra and eigenmodes, we derive two shape signatures. One that measures the similarity of points on a surface, and another that can be used to identify features of surfaces.

Fast Implementation of FDTD-Compatible Green's Function on Multicore Processor

In this letter, numerically efficient implementation of the finite-difference time domain (FDTD)-compatible Green's function on a multicore processor is presented. Recently, closed-form expression of this discrete Green's function (DGF) was derived, which simplifies its application in the FDTD simulations of radiation and scattering problems. Unfortunately, the new DGF expression involves binomial coefficients, whose computations may cause long runtimes and numerical problems. The proposed fast implementation of the DGF is based on the multiple precision arithmetic and employs a common programming language extended with the OpenMP parallel programming interface. As a result, the speedup factor of three orders of magnitude compared to the previous implementation was obtained, thus applicability of the DGF in FDTD simulations was significantly improved.

Implementation of FDTD-Compatible Green's Function on Graphics Processing Unit

In this letter, implementation of the finite-difference time domain (FDTD)-compatible Green's function on a graphics processing unit (GPU) is presented. Recently, closed-form expression for this discrete Green's function (DGF) was derived, which facilitates its applications in the FDTD simulations of radiation and scattering problems. Unfortunately, implementation of the new DGF formula in software requires a multiple precision arithmetic and may cause long runtimes. Therefore, we propose an acceleration of the DGF computations on a GPU employing the multiple precision arithmetic and the CUDA technology. We have achieved sixfold speedup relative to the reference DGF code executed on a multicore central processing unit. Results indicate that GPUs also represent an inexpensive source of computational power for accelerated DGF computations requiring the multiple precision arithmetic.

On infinity of the discrete spectrum of operators in the Friedrichs model

The discrete spectrumof selfadjoint operators in the Friedrichs model is studied. Necessary and sufficient conditions of existence of infinitely many eigenvalues in the Friedrichs model are presented. A discrete spectrum of a model three-particle discrete Schrodinger operator is described.

Multi-linear Formulation of Differential Geometry and Matris Regularizations

We prove that many aspects of the differential geometry of embedded Riemannian manifolds can be formulated in terms of multi-linear algebraic structures on the space of smooth functions. In particular, we find algebraic expressions for Weingartens formula, the Ricci curvature, and the Codazzi-Mainardi equations. For matrix analogues of embedded surfaces, we define discrete curvatures and Euler characteristics, and a non-commutative Gauss-Bonnet theorem is shown to follow. We derive simple expressions for the discrete Gauss curvature in terms of matrices representing the embedding coordinates, and explicit examples are provided. Furthermore, we illustrate the fact that techniques from differential geometry can carry over to matrix analogues by proving that a bound on the discrete Gauss curvature implies a bound on the eigenvalues of the discrete Laplace operator.

Rigidity of polyhedral surfaces, III

This paper investigates several global rigidity issues for polyhedral surfaces including inversive distance circle packings. Inversive distance circle packings are polyhedral surfaces introduced by P. Bowers and K. Stephenson as a generalization of Andreev-Thurston's circle packing. They conjectured that inversive distance circle packings are rigid. Using a recent work of R. Guo on variational principle associated to the inversive distance circle packing, we prove rigidity conjecture of Bowers-Stephenson in this paper. We also show that each polyhedral metric on a triangulated surface is determined by various discrete curvatures introduced in our previous work, verifying a conjecture in \cite{Lu1}. As a consequence, we show that the discrete Laplacian operator determines a Euclidean polyhedral metric up to scaling.

Extremal Theory for Spectrum of Random Discrete Schrödinger Operator. II. Distributions with Heavy Tails

We study the asymptotic structure of the first K largest eigenvalues λk,V and the corresponding eigenfunctions ψ(⋅;λk,V) of a finite-volume Anderson model (discrete Schrodinger operator) \(\mathcal{H}_{V}= \kappa \Delta_{V}+\xi(\cdot)\) on the multidimensional lattice torus V increasing to the whole of lattice ℤν, provided the distribution function F(⋅) of i.i.d. potential ξ(⋅) satisfies condition −log(1−F(t))=o(t3) and some additional regularity conditions as t→∞. For z∈V, denote by λ0(z) the principal eigenvalue of the “single-peak” Hamiltonian κΔV+ξ(z)δz in l2(V), and let \(\lambda^{0}_{k,V}\) be the kth largest value of the sample λ0(⋅) in V. We first show that the eigenvalues λk,V are asymptotically close to \(\lambda^{0}_{k,V}\). We then prove extremal type limit theorems (i.e., Poisson statistics) for the normalized eigenvalues (λk,V−BV)aV, where the normalizing constants aV>0 and BV are chosen the same as in the corresponding limit theorems for \(\lambda^{0}_{k,V}\). The eigenfunction ψ(⋅;λk,V) is shown to be asymptotically completely localized (as V↑ℤ) at the sites zk,V∈V defined by \(\lambda^{0}(z_{k,V})=\lambda^{0}_{k,V}\). Proofs are based on the finite-rank (in particular, rank one) perturbation arguments for discrete Schrodinger operator when potential peaks are sparse.

Uniform stability estimates for the discrete Calderón problems

In this paper, we focus on the analysis of discrete versions of the Calderón problem in dimension d ⩾ 3. In particular, our goal is to obtain stability estimates for the discrete Calderón problems that hold uniformly with respect to the discretization parameter. Our approach mimics the one in the continuous setting. Namely, we prove discrete Carleman estimates for the discrete Laplace operator. A main difference with the continuous ones is that in our case, the Carleman parameters cannot be taken arbitrarily large, but should be smaller than some frequency scale depending on the mesh size. Following the by-now classical complex geometric optics (CGO) approach, we can thus derive discrete CGO solutions, but with a limited range of parameters. As in the continuous case, we then use these solutions to obtain uniform stability estimates for the discrete Calderón problems.

Extinction and positivity of the solutions of the heat equations with absorption on networks

In this paper, we propose a discrete version of the following semilinear heat equation with absorption u t = Δ u − u q with q > 1 , which is said to be the ω- heat equation with absorption on a network. Using the discrete Laplacian operator Δ ω on a weighted graph, we define the ω-heat equations with absorption on networks and give their physical interpretations. The main concern is to investigate the large time behaviors of nontrivial solutions of the equations whose initial data are nonnegative and the boundary data vanish. It is proved that the asymptotic behaviors of the solutions u ( x , t ) as t tends to +∞ strongly depend on the sign of q − 1 .

Curvature, Geometry and Spectral Properties of Planar Graphs

We introduce a curvature function for planar graphs to study the connection between the curvature and the geometric and spectral properties of the graph. We show that non-positive curvature implies that the graph is infinite and locally similar to a tessellation. We use this to extend several results known for tessellations to general planar graphs. For non-positive curvature, we show that the graph admits no cut locus and we give a description of the boundary structure of distance balls. For negative curvature, we prove that the interiors of minimal bigons are empty and derive explicit bounds for the growth of distance balls and Cheeger’s constant. The latter are used to obtain lower bounds for the bottom of the spectrum of the discrete Laplace operator. Moreover, we give a characterization for triviality of the essential spectrum by uniform decrease of the curvature. Finally, we show that non-positive curvature implies the absence of finitely supported eigenfunctions for nearest neighbor operators.

Ill‐conditioning of the truncated singular value decomposition, Tikhonov regularization and their applications to numerical partial differential equations

AbstractThis paper explores some intrinsic characteristics of accuracy and stability for the truncated singular value decomposition (TSVD) and the Tikhonov regularization (TR), which can be applied to numerical solutions of partial differential equations (numerical PDE). The ill‐conditioning is a severe issue for numerical methods, in particular when the minimal singular value sigmamin of the stiffness matrix is close to zero, and when the singular vector umin of σmin is highly oscillating. TSVD and TR can be used as numerical techniques for seeking stable solutions of linear algebraic equations. In this paper, new bounds are derived for the condition number and the effective condition number which can be used to improve ill‐conditioning by TSVD and TR. A brief error analysis of TSVD and TR is also made, since both errors and condition number are essential for the numerical solution of PDE. Numerical experiments are reported for the discrete Laplace operator by the method of fundamental solutions. Copyright © 2011 John Wiley & Sons, Ltd.

Dirichlet forms for singular diffusion on graphs

We describe operators driving the time evolution of singular diffusion on finite graphs whose vertices are allowed to carry masses. The operators are defined by the method of quadratic forms on suitable Hilbert spaces. The model also covers quantum graphs and discrete Laplace operators. Mathematics subject classification (2010): 47D06, 60J60, 47E05, 35Q99, 05C99.

Approximation of the biharmonic problem using P1 finite elements

We study in this paper a P1 finite element approximation of the solution in of a biharmonic problem. Since the P1 finite element method only leads to an approximate solution in , a discrete Laplace operator is used in the numerical scheme. The convergence of the method is shown, for the general case of a solution with regularity, thanks to compactness results and to the use of a particular interpolation of regular functions with compact supports. An error estimate is proved in the case where the solution is in . The order of this error estimate is equal to 1 if the solution has a compact support, and only 1/5 otherwise. Numerical results show that these orders are not sharp in particular situations.

Radically elementary analysis of an interacting particle system at an unstable equilibrium

We investigate an interacting particle system consisting of two types of particles located at a finite point-lattice. The particles randomly change their type and neighboring particles randomly interchange positions. The system seems to remain at equilibrium for a substantial amount of time until it suddenly, at a critical time T , leaves equilibrium along what seems to be a deterministic trajectory. The analysis reveals, however, that the trajectories are determined randomly, but only by the systems behavior at very early times, much prior to T. In the nonstandard model used, the system randomly 'chooses' the trajectory in an infinitesimal interval (0;), 0, but this choice only becomes visible in the interval (T ; T). The underlying reason for this behavior is revealed by a decomposition of the systems trajectories with respect to an eigenbasis (gk)k2K of the discrete Laplace operator 4. It shows that after an initial random period the system's dynamics behaves, coordinate-wise, like t 7! e ( + k)(t T) k(!), where is unlimited ('infinitely large'), kgk =4gk and k(!) denotes a random quantity. The hyperfinite result obtained is translated into a standard limit theorem.

Voronoi-cell finite difference method for accurate electronic structure calculation of polyatomic molecules on unstructured grids

We introduce a new numerical grid-based method on unstructured grids in the three-dimensional real-space to investigate the electronic structure of polyatomic molecules. The Voronoi-cell finite difference (VFD) method realizes a discrete Laplacian operator based on Voronoi cells and their natural neighbors, featuring high adaptivity and simplicity. To resolve multicenter Coulomb singularity in all-electron calculations of polyatomic molecules, this method utilizes highly adaptive molecular grids which consist of spherical atomic grids. It provides accurate and efficient solutions for the Schrödinger equation and the Poisson equation with the all-electron Coulomb potentials regardless of the coordinate system and the molecular symmetry. For numerical examples, we assess accuracy of the VFD method for electronic structures of one-electron polyatomic systems, and apply the method to the density-functional theory for many-electron polyatomic molecules.

Parallel Poisson disk sampling with spectrum analysis on surfaces

The ability to place surface samples with Poisson disk distribution can benefit a variety of graphics applications. Such a distribution satisfies the blue noise property, i.e. lack of low frequency noise and structural bias in the Fourier power spectrum. While many techniques are available for sampling the plane, challenges remain for sampling arbitrary surfaces. In this paper, we present new methods for Poisson disk sampling with spectrum analysis on arbitrary manifold surfaces. Our first contribution is a parallel dart throwing algorithm that generates high-quality surface samples at interactive rates. It is flexible and can be extended to adaptive sampling given a user-specified radius field. Our second contribution is a new method for analyzing the spectral quality of surface samples. Using the spectral mesh basis derived from the discrete mesh Laplacian operator, we extend standard concepts in power spectrum analysis such as radial means and anisotropy to arbitrary manifold surfaces. This provides a way to directly evaluate the spectral distribution quality of surface samples without requiring mesh parameterization. Finally, we implement our Poisson disk sampling algorithm on the GPU, and demonstrate practical applications involving interactive sampling and texturing on arbitrary surfaces.