surface states and spectra let d 1 let h 0 be the discrete laplacian on the hilbert space l 2 d 1 with a dirichlet boundary condition and let v be a potential supported on the boundary d 1 we introduce the notions of surface states and surface spectrum of the operator h h 0 v and explore their properties our main result is that if the potential v is random and if the disorder is either large or small enough then in dimension two h has no surface spectrum on h 0 with probability one to prove this result we combine aizenman molchanov theory with techniques of scattering theory

maximum norm a posteriori error estimates for singularly perturbed reaction diffusion problems on anisotropic meshes residual type a posteriori error estimates in the maximum norm are given for singularly perturbed semilinear reaction diffusion equations posed in polygonal domains linear finite elements are considered on anisotropic triangulations the error constants are independent of the diameters and the aspect ratios of mesh elements and of the small perturbation parameter

anderson localization and lifshits tails for random surface potentials we consider schrodinger operators on l2 rd with a random potential concentrated near the surface rd1 0 rd we prove that the integrated density of states of such operators exhibits lifshits tails near the bottom of the spectrum from this and the multiscale analysis by boutet de monvel and stollmann arch math 80 2003 87 97 we infer anderson localization pure point spectrum and dynamical localization for low energies our proof of lifshits tails relies on spectral properties of schrodinger operators with partially periodic potentials in particular we show that the lowest energy band of such operators is parabolic

on a posteriori error estimator for primal equilibrium and mixed approximation of diffusion equations in this paper we present an a posteriori error estimator for diffusion equation with the solution obtained by black box solver the estimator is valid for classical primal equilibrium and mixed finite element approximations with or without numerical integration

a pde approach to fractional diffusion we derive a computable a posteriori error estimator for the harmonic extension problem which localizes the fractional powers of elliptic operators supplemented with dirichlet boundary conditions our a posteriori error estimator relies on the solution of small discrete problems on anisotropic cylindrical stars it exhibits built in flux equilibration and is equivalent to the energy error up to data oscillation under suitable assumptions we design a simple adaptive algorithm and present numerical experiments which reveal a competitive performance

metric tensors for anisotropic mesh generation it has been amply demonstrated that significant improvements in accuracy and efficiency can be gained when a properly chosen anisotropic mesh is used in the numerical solution for a large class of problems which exhibit anisotropic solution features in practice an anisotropic mesh is commonly generated as a quasi uniform mesh in the metric determined by a tensor specifying the shape size orientation of elements thus it is crucial to choose an appropriate metric tensor for anisotropic mesh generation and adaptation in this paper we develop a general formula for the metric tensor for use in any spatial dimension the formulation is based on error estimates for polynomial preserving interpolation on simiplicial elements numerical results in two dimensions are presented to demonstrate the ability of the metric tensor to produce anisotropic meshes with correct mesh concentration and good overall quality the procedure developed in this paper for defining the metric tensor can also be applied to other types of error estimates

localization at large disorder and at extreme energies an elementary derivation the work presents a short proof of localization under the conditions of either strong disorder 0 or extreme energies for a wide class of self adjoint operators with random matrix elements acting inl 2 spaces a prototypical example is the discrete schrodinger operatorh u 0 x v x onz d d 1 withu 0 x a specified background potential and v x generated as random variables the general results apply to operators with replaced by a non local self adjoint operatort whose matrix elements satisfy y t x y s const uniformly inx for somes 1 localization means here that within a specified energy range the spectrum ofh is of the pure point type or equivalently the wave functions do not spread indefinitely under the unitary time evolution generated byh the effect is produced by strong disorder in either the potential or in the off diagonal matrix elementst x y under rapid decay oft x y the corresponding eigenfunctions are also proven to decay exponentially the method is based on resolvent techniques the central technical ideas include the use of low moments of the resolvent kernel i e g e x y s withs small enough 1 to avoid the divergence caused by the distribution s cauchy tails and an effective use of the simple form of the dependence ofg e x y on the individual matrix elements ofh in elucidating the implications of the fundamental equation h e g e x x 0 x x0 this approach simplifies previous derivations of localization results avoiding the small denominator difficulties which have been hitherto encountered in the subject it also yields some new results which include localization under the following sets of conditions i potentials with an inhomogeneous non random partu 0 x ii the bethe lattice iii operators with very slow decay in the off diagonal terms t x y 1 x y d e and iv localization produced by disordered boundary conditions

robust a posteriori error estimators for a singularly perturbed reaction diffusion equation we derive robust a posteriori error estimators for a singularly perturbed reaction diffusion equation here robust means that the estimators yield global upper and local lower bounds on the error measured in the energy norm such that the ratio of the upper and lower bounds is bounded from below and from above by constants which do neither depend on any meshsize nor on the perturbation parameter the estimators are based either on the evaluation of local residuals or on the solution of discrete local dirichlet or neumann problems

an adaptive algorithm for the crank nicolson scheme applied to a time dependent convection diffusion problem an a posteriori upper bound is derived for the nonstationary convection diffusion problem using the crank nicolson scheme and continuous piecewise linear stabilized finite elements with large aspect ratio following lozinski et al 2009 13 a quadratic time reconstruction is used a space and time adaptive algorithm is developed to ensure the control of the relative error in the l 2 h 1 norm numerical experiments illustrating the efficiency of this approach are reported it is shown that the error indicator is of optimal order with respect to both the mesh size and the time step even in the convection dominated regime and in the presence of boundary layers

many body localization in the droplet spectrum of the random xxz quantum spin chain abstract we study many body localization properties of the disordered xxz spin chain in the ising phase disorder is introduced via a random magnetic field in the z direction we prove a strong form of dynamical exponential clustering for eigenstates in the droplet spectrum for any pair of local observables separated by a distance l the sum of the associated correlators over these states decays exponentially in l in expectation this exponential clustering persists under the time evolution in the droplet spectrum our result applies to the large disorder regime as well as to the strong ising phase at fixed disorder with bounds independent of the support of the observables

localization of surface spectra we study spectral properties of the discrete laplacian h on the half space with random boundary condition the v n are independent random variables on a probability space and is the coupling constant it is known that if the v n have densities then on the interval 2 d 1 2 d 1 h 0 the spectrum of the dirichlet laplacian the spectrum of h is p a s absolutely continuous for all jl1 here we show that if the random potential p satisfies the assumption of aizenman molchanov am then there are constants d and d such that for d the spectrum of h outside h 0 is p a s pure point with exponentially decaying eigenfunctions

the sharp sobolev type inequalities in the lorentz sobolev spaces in the hyperbolic spaces abstract let w 1 l p q h n 1 q p denote the lorentz sobolev spaces of order one in the hyperbolic spaces h n our aim in this paper is three fold first of all we establish a sharp poincare inequality in w 1 l p q h n with 1 q p which generalizes the result in 41 to the setting of lorentz sobolev spaces second we prove several sharp poincare sobolev type inequalities in w 1 l p q h n with 1 q p n which generalize the results in 45 to the setting of lorentz sobolev spaces finally we provide the improved moser trudinger type inequalities in w 1 l n q h n in the critical case p n with 1 q n which generalize the results in 43 and improve the results in 57 in the proof of the main results we shall prove a polya szego type principle in w 1 l p q h n with 1 q p which maybe is of independent interest

spectral structure of anderson type hamiltonians we study self adjoint operators of the form h h 0 n n where the n s are a family of orthonormal vectors and the n s are independently distributed random variables with absolutely continuous probability distributions we prove a general structural theorem saying that for each pair n m if the cyclic subspaces corresponding to the vectors n and m are not completely orthogonal then the restrictions of h to these subspaces are unitarily equivalent with probability one this has some consequences for the spectral theory of such operators in particular we show that well behaved absolutely continuous spectrum of anderson type hamiltonians must be pure and use this to prove the purity of absolutely continuous spectrum in some concrete cases

regularity of the surface density of states abstract we prove that the integrated surface density of states of continuous or discrete anderson type random schrodinger operators is a measurable locally integrable function rather than a signed measure or a distribution this generalizes our recent results on the existence of the integrated surface density of states in the continuous case and those of a chahrour in the discrete case the proof uses the new l p bound on the spectral shift function recently obtained by combes hislop and nakamura also we provide a simple proof of their result on the holder continuity of the integrated density of bulk states

wave operators for the surface maryland model we study scattering properties of the discrete laplacian h on the half space z d 1 zd z with the boundary condition n 1 tan n n 0 where 0 1 d we denote by h0 the dirichlet laplacian on z d 1 khoruzenko and pastur phys rep 288 109 126 1997 have shown that if has typical diophantine properties then the spectrum of h on r h0 is pure point and that corresponding eigenfunctions decay exponentially we demonstrated in an earlier paper lett math phys 45 185 1998 that for every independent over the rationals the spectrum of h on h0 is purely absolutely continuous in this paper we continue the analysis of h on h0 and prove that whenever is independent over the rationals the wave operators h h0 exist and are complete on h0 moreover we show that under the same conditions h has no surface states on h0

anderson localization and the space time characteristic of continuum states a proof of anderson localization is obtained by ruling out any continuous spectrum on the basis of the space time characteristic of its states

optimal anisotropic meshes for minimizing interpolation errors in l p norm in this paper we present a new optimal interpolation error estimate in l p norm 1 p for finite element simplicial meshes in any spatial dimension a sufficient condition for a mesh to be nearly optimal is that it is quasi uniform under a new metric defined by a modified hessian matrix of the function to be interpolated we also give new functionals for the global moving mesh method and obtain optimal monitor functions from the viewpoint of minimizing interpolation error in the l p norm some numerical examples are also given to support the theoretical estimates

edge residuals dominate a posteriori error estimates for linear finite element methods on anisotropic triangular and tetrahedral meshes summary both for the h 1 and l 2 norms we prove that up to higher order perturbation terms edge residuals yield global upper and local lower bounds on the error of linear finite element methods on anisotropic triangular or tetrahedral meshes we also show that with a correct scaling edge residuals yield a robust error estimator for a singularly perturbed reaction diffusion equation

cantor spectrum of graphene in magnetic fields this research was partially supported by the nsf dms 1401204 support by the epsrc grant ep l016516 1 for the university of cambridge cdt the cca is gratefully acknowledged s b r h would like to thank the institute for advanced study princeton for its hospitality during the 2017 18 academic year r h was supported in part by nsf dms 1638352

anisotropic residual based a posteriori mesh adaptation in 2d element based approach abstract an element based adaptation method is developed for an anisotropic a posteriori error estimator the adaptation does not make use of a metric but instead equidistributes the error over elements using local mesh modifications numerical results are reported comparing with three popular anisotropic adaptation methods currently in use it was found that the new method gives favourable results for controlling the energy norm of the error in terms of degrees of freedom at the cost of increased cpu usage additionally we considered a new l 2 variant of the estimator the estimator is shown to be conditionally equivalent to the exact l 2 error we provide examples of adapted meshes with the l 2 estimator and show that it gives greater control of the l 2 error compared with the original estimator