We review recent advances in the spectral theory of Schrödinger operators with decaying potentials. The area has seen spectacular progress in the past few years, stimulated by several conjectures stated by Barry Simon starting at the 1994 International Congress on Mathematical Physics in Paris. The one-dimensional picture is now fairly complete, and provides many striking spectral examples. The...

We present in this paper the convergence properties of Jacobi spectral collocation method when used to approximate the solution of multidimensional nonlinear Volterra integral equation. The solution is sufficiently smooth while the source function and the kernel function are smooth. We choose the Jacobi-Gauss points associated with the multidimensional Jacobi weight function [Formula: see text]...

This paper presents the formulation, validation, and application of a spectral/hp algorithm for numerical solution of the Navier–Stokes and heat/scalar transport equations in arbitrarily moving complex geometries. The new spectral element algorithm is based on the arbitrary Lagrangian Eulerian formulation and utilizes modal expansion of Jacobi polynomials in mixed triangular and quadrilateral e...

Algebraic multigrid is investigated as a solver for linear systems that arise from high-order spectral element discretizations. An algorithm is introduced that utilizes the efficiency of low-order finite elements to precondition the high-order method in a multilevel setting. In particular, the efficacy of this approach is highlighted on simplexes in two and three dimensions with nodal spectral ...

The density of states and the optical conductivity of the doped Hubbard model on a Bethe lattice with infinite connectivities have been studied using an analytic variant of the Lanczos continued fraction method. The spectral weight of the gap states and the position of the chemical potential upon hole or electron doping have been studied. We argue that the strong correlation effects such as gap...

As the scaling of CMOS transistors extends to the sub-20 nm regime, the most challenging aspect of device design is the control of the off-state current. The traditional methods for controlling leakage current via the substrate doping profile will be difficult to implement at these dimensions. A promising method for controlling leakage in sub-20 nm transistors is the reduction in source-to-drai...

There is mounting observational evidence that the expansion of our universe is undergoing an acceleration. A dark energy component has usually been invoked as the most feasible mechanism for the acceleration. However, it is desirable to explore alternative possibilities motivated by particle physics before adopting such an untested entity. In this work, we focus our attention on an acceleration...

The zero temperature Mott-Hubbard transition as a function of the Coulomb repulsion U is investigated in the limit of large dimensions. The behavior of the density of states near the transition at U ­ Uc is analyzed in all orders of the skeleton expansion. It is shown that only two transition scenarios are consistent with the skeleton expansion for U , Uc: (i) The Mott-Hubbard transition is “di...

We consider the problem of randomly distributed positive delta-function scatterers in a strong magnetic field and study the behavior of density of states close to the spectral boundary at E = h̄ωc/2 in both two and three dimensions. Starting from dimensionally reduced expression of Brezin et al. and using the semiclassical approximation we show that the density of states in the Lifshitz tail at ...

This text is a survey of recent results obtained by the author and collaborators on different problems for non-self-adjoint operators. The topics are: Kramers-Fokker-Planck type operators, spectral asymptotics in two dimensions and Weyl asymptotics for the eigenvalues of non-self-adjoint operators with small random perturbations. In the introduction we also review the notion of pseudo-spectrum ...