Discrete Schrodinger Operators Research Articles

Transient analysis of finite multi‐conductor transmission line metamaterials (MTL‐MMs) based on discrete green's function (DGF) and macro‐modelling techniques

A new systematic macro-modelling technique for transient analysis of finite multi-conductor transmission line metamaterials (MTL-MMs) based on a Dyadic discrete Green's function (DGF) approach is proposed. Governing discrete voltage-based equations for one- and two-dimensional (1-D and 2-D) MTL-MMs are derived using a rigorous MTL analysis. Applying the idea of the Dyadic DGF solution to these equations, the impedance matrix representation of a finite open-ended MTL-MM is represented in a rational form, where the corresponding poles and residues can be identified exactly. The resulting pole/residue macro-model is converted into a state space model which is compatible to SPICE circuit simulator. The proposed macro-modelling technique significantly reduces the CPU time for transient analysis of 2-D electromagnetic band gap (EBG) structures embedded in large multi-layer printed circuit boards or volumetric NRI slabs with interaction with free space waves which can be modelled as an MTL-MM system. The usefulness of the proposed macro-model is illustrated by three sample MTL-MMs including 1-D shunt node negative refractive index ( NRI ) slab and 2-D two- and three-layered shielded EBG structures. The obtained results from the proposed macro-modelling technique are presented and compared with those obtained using a full-wave simulator, confirming the validity of the proposed model.

Homogenization theory for the random conductance model with degenerate ergodic weights and unbounded-range jumps

Nous considerons les proprietes d’homogeneisation de l’operateur de Laplace discret avec des conductances aleatoires. Nous demontrons l’homogeneisation de l’equation de Poisson discrete et des plus hauts elements du spectre de l’operateur de Dirichlet dans un domaine limite. Nous supposons les conductances stationnaires, ergodiques et strictement positives a plus proches voisins. Compare aux resultats precedents, nous remplacons l’ellipticite uniforme par des conditions d’integrabilite des moments des conductances. De plus, nous autorisons des sauts de tailles arbitraires. En l’absence de sauts longs, les conditions sur les moments sont optimales pour l’homogeneisation spectrale. Elles correspondent a la condition necessaire du theoreme central limite pour les marches aleatoires en conductances aleatoires. Nous utilisons l’homogeneisation spectrale pour demontrer un principe de grandes deviations gele pour le temps local normalise de la marche aleatoire dans une suite croissante de boites. Nos demonstrations sont basees sur un resultat de compacite pour l’energie de Dirichlet, les inegalites de Poincare, l’iteration de Moser et la convergence a deux echelles.

Optimal Convergence Analysis of a Second Order Scheme for a Thin Film Model Without Slope Selection

In Li et al. (J Sci Comput 76(3):1905–1937, 2018), a temporal second-order mixed finite element scheme has been proposed for the thin film epitaxial growth model without slope selection. Using the super-convergence theory in a regular rectangular mesh, the authors of Li et al. (2018) proved an optimal $$O(h^{q+1}+\tau ^2)$$ convergence. However, in a quasi-uniform triangulation mesh setting, only a sub-optimal convergence rate $$O(h^q+\tau ^2)$$ is proved, while numerical results indicated an optimal $$O(h^{q+1}+\tau ^2)$$ convergence when the exact solution has $$H^{q+1}$$ regularity in space. Here h and $$\tau $$ are the discretization sizes in space and time, respectively, and $$q\ge 1$$ is the degree of the polynomial in the spatial discretization. In this paper, we provide a theoretical proof of the optimal convergence rate. The main difficulty lies in how to treat a nonlinear term $$\frac{\nabla u}{1+|\nabla u|^2}$$ . We solve this by using a discrete Laplacian operator $$-\varDelta _h$$ and some uncommon techniques in the analysis. Numerical results are also presented to demonstrate the $$(q+1)$$ -order convergence of the spatial approximation.

Fundamental solutions for discrete dynamical systems involving the fractional Laplacian

We prove representation results for solutions of a time‐fractional differential equation involving the discrete fractional Laplace operator in terms of generalized Wright functions. Such equations arise in the modeling of many physical systems, for example, chain processes in chemistry and radioactivity. Our focus is in the problem , where 0<β ≤ 2, 0<α ≤ 1, , (−Δd)α is the discrete fractional Laplacian, and is the Caputo fractional derivative of order β. We discuss important special cases as consequences of the representations obtained.

A Fast Poisson Solver of Second-order Accuracy for Isolated Systems in Three-dimensional Cartesian and Cylindrical Coordinates

Abstract We present an accurate and efficient method to calculate the gravitational potential of an isolated system in 3D Cartesian and cylindrical coordinates subject to vacuum (open) boundary conditions. Our method consists of two parts: an interior solver and a boundary solver. The interior solver adopts an eigenfunction expansion method together with a tridiagonal matrix solver to solve the Poisson equation subject to the zero boundary condition. The boundary solver employs James’s method to calculate the boundary potential due to the screening charges required to keep the zero boundary condition for the interior solver. A full computation of gravitational potential requires running the interior solver twice and the boundary solver once. We develop a method to compute the discrete Green’s function in cylindrical coordinates, which is an integral part of the James algorithm to maintain second-order accuracy. We implement our method in the Athena++ magnetohydrodynamics code and perform various tests to check that our solver is second-order accurate and exhibits good parallel performance.

Regularity and backward compactness of attractors for non-autonomous lattice systems with random coefficients

We study longtime behavior for the non-autonomous lattice model with multiplicative white noise and a random coefficient in the discrete Laplace operator. We first show existence of a bi-spatial attractor when the initial space is the weighted square summation space and the terminal space is the weighted p-times summation space for p > 2. We then show backward compactness of the attractor in both initial and terminal spaces if the time-indexed forces are backward-tempered and backward-null. Finally, by proving identity of the attractors on the different universes of tempered or backward tempered sets, we show measurability of the attractor in the initial space and in the terminal space, respectively.

Long-Range Scattering for Discrete Schrödinger Operators

In this paper, we define time-independent modifiers to construct a long-range scattering theory for a class of difference operators on $$\mathbb {Z}^d$$ , including the discrete Schrodinger operators on the square lattice. The modifiers are constructed by observing the corresponding Hamilton flow on $$T^*\mathbb {T}^d$$ . We prove the existence and completeness of modified wave operators in terms of the above-mentioned time-independent modifiers.

Recurrence scheme for FDTD-compatible discrete Green's function derived based on properties of Gauss hypergeometric function

ABSTRACTIn this paper, the formulation of one-dimensional FDTD (Finite-difference time-domain)-compatible discrete Green's function (DGF) is derived based on the Gauss hypergeometric function (GHF). The properties of GHF make it possible to derive the recurrence scheme only in the time domain for the DGF generation. Furthermore, this recurrence scheme is valid for any stable time-step size and can be implemented using standard numerical precision of computations. The proposed derivation is obtained without processing in any symbolic mathematics software and relies on the application of known properties of GHF. The difference between the developed recurrence scheme and the direct FDTD simulation is approximately at the level of numerical noise, which confirms the correctness of our derivation. The results obtained should be useful for the development of computational techniques employing FDTD and the diakoptic approach.

Computation of volume potentials on structured grids with the method of local corrections

We present a new version of the Method of Local Corrections (MLC) \cite{mlc}, a multilevel, low communications, non-iterative, domain decomposition algorithm for the numerical solution of the free space Poisson's equation in 3D on locally-structured grids. In this method, the field is computed as a linear superposition of local fields induced by charges on rectangular patches of size $O(1)$ mesh points, with the global coupling represented by a coarse grid solution using a right-hand side computed from the local solutions. In the present method, the local convolutions are further decomposed into a short-range contribution computed by convolution with the discrete Green's function for an $Q^{th}$-order accurate finite difference approximation to the Laplacian with the full right-hand side on the patch, combined with a longer-range component that is the field induced by the terms up to order $P-1$ of the Legendre expansion of the charge over the patch. This leads to a method with a solution error that has an asymptotic bound of $O(h^P) + O(h^Q) + O(\epsilon h^2) + O(\epsilon)$, where $h$ is the mesh spacing, and $\epsilon$ is the max norm of the charge times a rapidly-decaying function of the radius of the support of the local solutions scaled by $h$. Thus we have eliminated the low-order accuracy of the original method (which corresponds to $P=1$ in the present method) for smooth solutions, while keeping the computational cost per patch nearly the same with that of the original method. Specifically, in addition to the local solves of the original method we only have to compute and communicate the expansion coefficients of local expansions (that is, for instance, 20 scalars per patch for $P=4$). Several numerical examples are presented to illustrate the new method and demonstrate its convergence properties.

Discrete Magnetic Bottles on Quasi-Linear Graphs

We study discrete magnetic Schrodinger operators on graphs that are obtained by adding decorations to the linear graph $$\mathbb {N}$$ . We construct examples in which the magnetic field destroys the essential spectrum. The methods rely on spectral comparison theorems for quadratic forms and a careful study of an explicit Schrodinger operator on a finite graph.

Weighted estimates for the Laplacian on the cubic lattice

We consider the discrete Laplacian $\Delta$ on the cubic lattice $\mathbb{Z}^d$, and deduce estimates on the group $e^{i t \Delta}$ and the resolvent $(\Delta-z)^{-1}$, weighted by $\ell^q (\mathbb{Z}^d)$-weights for suitable $q \geqslant 2$. We apply the obtained results to discrete Schrodinger operators in dimension $d \geqslant 3$ with potentials from $\ell^p (\mathbb{Z}^d)$ with suitable $p \geqslant1$.

Dynamic inverse problem for Jacobi matrices

We consider the inverse dynamical problem for the dynamical system with discrete time associated with the semi-infinite Jacobi matrix. We solve the inverse problem for such a system and answer a question on the characterization of the inverse data. As a by-product we give a necessary and sufficient condition for the measure on the real line line to be the spectral measure of semi-infinite discrete Schrodinger operator.

Titchmarsh–Weyl theory for vector-valued discrete Schrödinger operators

We develop the Titchmarsh–Weyl theory for vector-valued discrete Schrodinger operators. We show that the Weyl m functions associated with these operators are matrix valued Herglotz functions that map complex upper half plane to the Siegel upper half space. We discuss about the Weyl disk and Weyl circle corresponding to these operators by defining these functions on a bounded interval. We also discuss the geometric properties of Weyl disk and find the center and radius of the Weyl disk explicitly in terms of matrices.

Anderson localization for radial tree graphs with random branching numbers

We prove Anderson localization for the discrete Laplace operator on radial tree graphs with random branching numbers. Our method relies on the representation of the Laplace operator as the direct sum of half-line Jacobi matrices whose entries are non-degenerate, independent, identically distributed random variables with singular distributions.

Gelfand–Yaglom formula for functional determinants in higher dimensions

The Gelfand–Yaglom formula relates functional determinants of the one-dimensional second order differential operators to the solutions of the corresponding initial value problem. In this work we generalise the Gelfand–Yaglom method by considering discrete and continuum partial second order differential operators in higher dimensions. To illustrate our main result we apply the generalised formula to the two-dimensional massive and massless discrete Laplace operators and calculate asymptotic expressions for their determinants.

Redesign of the Laplacian kernel for improvements in conductivity imaging using MRI.

To develop an electrical property tomography reconstruction method that achieves improvements over standard method by redesigning the Laplacian kernel. A decomposition property of the governing PET equation shows the possibility of redesigning the Laplacian kernel for conductivity reconstruction. Hence, the discrete Laplacian operator used for electrical property tomography reconstruction is redesigned to have a Gaussian-like envelope, which enables manipulation of the spatial and spectral response. The characteristics of the proposed kernel are investigated through numerical simulations and in vivo brain experiments. The proposed method reduces textured noise, which hampers observing features of the conductivity image. Furthermore, the proposed scheme can mitigate the propagation of local phase error such as flow-induced phase. By doing so, the proposed method can recover feature information in conductivity (or resistivity) images. Lastly, the proposed kernel can be extended to other electrical property tomography reconstructions, improving the quality of images. An alternative design of the Laplacian kernel for conductivity imaging has been developed to mitigate the textured noise and the propagation of local phase artifact.

Error Analysis of a Fully Discrete Morley Finite Element Approximation for the Cahn–Hilliard Equation

This paper proposes and analyzes the Morley element method for the Cahn–Hilliard equation. It is a fourth order nonlinear singular perturbation equation arises from the binary alloy problem in materials science, and its limit is proved to approach the Hele-Shaw flow. If the $$L^2(\Omega )$$ error estimate is considered directly as in paper [14], we can only prove that the error bound depends on the exponential function of $$\frac{1}{\epsilon }$$ . Instead, this paper derives the error bound which depends on the polynomial function of $$\frac{1}{\epsilon }$$ by considering the discrete $$H^{-1}$$ error estimate first. There are two main difficulties in proving this polynomial dependence of the discrete $$H^{-1}$$ error estimate. Firstly, it is difficult to prove discrete energy law and discrete stability results due to the complex structure of the bilinear form of the Morley element discretization. This paper overcomes this difficulty by defining four types of discrete inverse Laplace operators and exploring the relations between these discrete inverse Laplace operators and continuous inverse Laplace operator. Each of these operators plays important roles, and their relations are crucial in proving the discrete energy law, discrete stability results and error estimates. Secondly, it is difficult to prove the discrete spectrum estimate in the Morley element space because the Morley element space intersects with the $$C^1$$ conforming finite element space but they are not contained in each other. Instead of proving this discrete spectrum estimate in the Morley element space, this paper proves a generalized coercivity result by exploring properties of the enriching operators and using the discrete spectrum estimate in its $$C^1$$ conforming relative finite element space, which can be obtained by using the spectrum estimate of the Cahn–Hilliard operator. The error estimate in this paper provides an approach to prove the convergence of the numerical interfaces of the Morley element method to the interface of the Hele-Shaw flow.

Engine-Scalable Rotor Casing Convective Heat Flux Evaluation Using Inverse Heat Transfer Methods

In this paper, we propose a two-step methodology to evaluate the convective heat flux along the rotor casing using an engine-scalable approach based on discrete Green's functions . The first step consists in the use of an inverse heat transfer technique to retrieve the heat flux distribution on the shroud inner wall by measuring the temperature of the outside wall; the second step is the calculation of the convective heat flux at engine conditions, using the experimental heat flux and the Green functions engine-scalable technique. Inverse methodologies allow the determination of boundary conditions; in this case, the inner casing surface heat flux, based on measurements from outside of the system, which prevents aerothermal distortion caused by routing the instrumentation into the test article. The heat flux, retrieved from the inverse heat transfer methodology, is related to the rotor tip gap. Therefore, for a given geometry and tip gap, the pressure and temperature can also be retrieved. In this work, the digital filter method is applied in order to take advantage of the response of the temperature to heat flux pulses. The discrete Green's function approach employs a matrix to relate an arbitrary temperature distribution to a series of pulses of heat flux. In this procedure, the terms of the Green's function matrix are evaluated with the output of the inverse heat transfer method. Given that key dimensionless numbers are conserved, the Green's functions matrix can be extrapolated to engine-like conditions. A validation of the methodology is performed by imposing different arbitrary heat flux distributions, to finally demonstrate the scalability of the Green's function method to engine conditions. A detailed uncertainty analysis of the two-step routine is included based on the value of the pulse of heat flux, the temperature measurement uncertainty, the thermal properties of the material, and the physical properties of the rotor casing.

Proof of dynamical localization for perturbations of discrete 1D Schrödinger operators with uniform electric fields

We present a proof of discrete spectrum and dynamical localization for small perturbations of discrete one-dimensional Schrodinger operators with uniform electric fields. The proof of dynamical localization is based on the KAM technique.

Sparse Approximation of 3D Meshes Using the Spectral Geometry of the Hamiltonian Operator

The discrete Laplace operator is ubiquitous in spectral shape analysis, since its eigenfunctions are provably optimal in representing smooth functions defined on the surface of the shape. Indeed, subspaces defined by its eigenfunctions have been utilized for shape compression, treating the coordinates as smooth functions defined on the given surface. However, surfaces of shapes in nature often contain geometric structures for which the general smoothness assumption may fail to hold. At the other end, some explicit mesh compression algorithms utilize the order by which vertices that represent the surface are traversed, a property which has been ignored in spectral approaches. Here, we incorporate the order of vertices into an operator that defines a novel spectral domain. We propose a method for representing 3D meshes using the spectral geometry of the Hamiltonian operator, integrated within a sparse approximation framework. We adapt the concept of a potential function from quantum physics and incorporate vertex ordering information into the potential, yielding a novel data-dependent operator. The potential function modifies the spectral geometry of the Laplacian to focus on regions with finer details of the given surface. By sparsely encoding the geometry of the shape using the proposed data-dependent basis, we improve compression performance compared to previous results that use the standard Laplacian basis and spectral graph wavelets.