spectral variation bounds in hyperbolic geometry abstract we derive new estimates for distances between optimal matchings of eigenvalues of non normal matrices in terms of the norm of their difference we introduce and estimate a hyperbolic metric analogue of the classical spectral variation distance the result yields a qualitatively new and simple characterization of the localization of eigenvalues our bound improves on the best classical spectral variation bounds due to krause if the distance of matrices is sufficiently small and is sharp for asymptotically large matrices our approach is based on the theory of model operators which provides us with strong resolvent estimates the latter naturally lead to a chebyshev type interpolation problem with finite blaschke products which can be solved explicitly and gives stronger bounds than the classical chebyshev interpolation with polynomials as compared to the classical approach our method does not rely on hadamard s inequality and immediately generalizes to algebraic operators on hilbert space

moment analysis for localization in random schrodinger operators we study localization effects of disorder on the spectral and dynamical properties of schrodinger operators with random potentials the new results include exponentially decaying bounds on the transition amplitude and related projection kernels including in the mean these are derived through the analysis of fractional moments of the resolvent which are finite due to the resonance diffusing effects of the disorder the main difficulty which has up to now prevented an extension of this method to the continuum can be traced to the lack of a uniform bound on the lifshitz krein spectral shift associated with the local potential terms the difficulty is avoided here through the use of a weak l 1 estimate concerning the boundary value distribution of resolvents of maximally dissipative operators combined with standard tools of relative compactness theory

infinite dimensional features of matrices and pseudospectra abstract given a hilbert space operator t the level sets of function t z t z 1 1 determine the so called pseudospectra of t we set t to be zero on the spectrum of t after giving some elementary properties of t which as it seems were not noticed before we apply them to the study of the approximation we prove that for any operator t there is a sequence t n of finite matrices such that t n z tends to t z uniformly on c in this proof quasitriangular operators play a special role this is merely an existence result we do not give a concrete construction of this sequence of matrices one of our main points is to show how to use infinite dimensional operator models in order to produce examples and counterexamples in the set of finite matrices of large size in particular we get a result which means in a sense that the pseudospectrum of a nilpotent matrix can be anything one can imagine we also study the norms of the multipliers in the context of cowen douglas class operators we use these results to show that to the opposite to the function s the function s z for certain finite matrices s may oscillate arbitrarily fast even far away from the spectrum

localization criteria for anderson models on locally finite graphs we prove spectral and dynamical localization for anderson models on locally finite graphs using the fractional moment method our theorems extend earlier results on localization for the anderson model on d we establish geometric assumptions for the underlying graph such that localization can be proven in the case of sufficiently large disorder

a note on fractional moments for the one dimensional continuum anderson model we give a proof of dynamical localization in the form of exponential decay of spatial correlations in the time evolution for the one dimensional continuum anderson model via the fractional moments method this follows via exponential decay of fractional moments of the green function which is shown to hold at arbitrary energy and for any single site distribution with bounded compactly supported density

harmonic analysis on sl 2 r and smoothness of the density of states in the one dimensional anderson model we consider infinite jacobi matrices with ones off diagonal and independent identically distributed random variables with distributionf v dv on diagonal iff has compact support and lies in some sobolev spacel 1 then we prove that the integrated density of states k e isc ine

maximum of the resolvent over matrices with given spectrum abstract in numerical analysis it is often necessary to estimate the condition number c n t t t 1 and the norm of the resolvent t 1 of a given n n matrix t we derive new spectral estimates for these quantities and compute explicit matrices that achieve our bounds we recover the fact that the supremum of c n t over all matrices with t 1 and minimal absolute eigenvalue r min t 0 is the kronecker bound 1 r n this result is subsequently generalized by computing for given in the closed unit disc the supremum of t 1 where t 1 and the spectrum t of t is constrained to remain at a pseudo hyperbolic distance of at least r 0 1 around we find that the supremum is attained by a triangular toeplitz matrix this provides a simple class of structured matrices on which condition numbers and resolvent norm bounds can be studied numerically the occurring toeplitz matrices are so called model matrices i e matrix representations of the compressed backward shift operator on the hardy space h 2 to a finite dimensional invariant subspace

discrete approximations and refined euler lagrange conditions fornonconvex differential inclusions this paper deals with the bolza problem p for differential inclusions subject to general endpoint constraints we pursue a twofold goal first we develop a finite difference method for studying p and construct a discrete approximation to p that ensures a strong convergence of optimal solutions second we use this direct method to obtain necessary optimality conditions in a refined euler lagrange form without standard convexity assumptions in general we prove necessary conditions for the so called intermediate relaxed local minimum that takes an intermediate place between the classical concepts of strong and weak minima in the case of a mayer cost functional or boundary solutions to differential inclusions this euler lagrange form holds without any relaxation the results obtained are expressed in terms of nonconvex valued generalized differentiation constructions for nonsmooth mappings and sets

dynamical localization for discrete anderson dirac operators we establish dynamical localization for random dirac operators on the d dimensional lattice with d in left 1 2 3 right in the three usual regimes large disorder band edge and 1d these operators are discrete versions of the continuous dirac operators and consist in the sum of a discrete free dirac operator with a random potential the potential is a diagonal matrix formed by different scalar potentials which are sequences of independent and identically distributed random variables according to an absolutely continuous probability measure with bounded density and of compact support we prove the exponential decay of fractional moments of the green function for such models in each of the above regimes i e j throughout the spectrum at larger disorder jj for energies near the band edges at arbitrary disorder and jjj in dimension one for all energies in the spectrum and arbitrary disorder dynamical localization in theses regimes follows from the fractional moments method the result in the one dimensional regime contrast with one that was previously obtained for 1d dirac model with bernoulli potential

optimization and controllability without differentiability assumptions let mathcal x and mathcal y be real normed vector spaces k subset mathcal x convex and compact c subset mathcal y a convex body mathcal u subset k and phi phi k to mathbb r m times mathcal y a function that can be appropriately approximated by functions phi i phi i whose compositions with linear maps of small finite dimensional simplices are c 1 we derive sufficient conditions for phi to be locally controllable on mathcal u subject to the restriction phi u in c and obtain as a corollary corresponding necessary conditions for a related restricted minimum these conditions are formulated in terms of directional derivate containers which are a type of set valued derivatives of phi phi and they improve on and extend previously obtained results they are used elsewhere to obtain new conditions for controllability and restricted minimum in nonsmooth optimal control problems defined by differential or functional integral equatio

anderson localization for bernoulli and other singular potentials we prove exponential localization in the anderson model under very weak assumptions on the potential distribution in one dimension we allow any measure which is not concentrated on a single point and possesses some finite moment in particular this solves the longstanding problem of localization for bernoulli potentials i e potentials that take only two values in dimensions greater than one we prove localization at high disorder for potentials with holder continuous distributions and for bounded potentials whose distribution is a convex combination of a holder continuous distribution with high disorder and an arbitrary distribution these include potentials with singular distributions we also show that for certain bernoulli potentials in one dimension the integrated density of states has a nontrivial singular component

absence of diffusion in the anderson tight binding model for large disorder or low energy we prove that the green s function of the anderson tight binding hamiltonian decays exponentially fast at long distances on v with probability 1 we must assume that either the disorder is large or the energy is sufficiently low our proof is based on perturbation theory about an infinite sequence of block hamiltonians and is related to kam methods

anderson localization for multi dimensional systems at large disorder or large energy we prove that discrete schrodinger operators on d with a random potential have almost surely only pure point spectrum and exponentially decaying eigenfunctions for large disorder or large energy this is the first proof of localization for multi dimensional anderson models

dynamical delocalization for the 1d bernoulli discrete dirac operator a 1d tight binding version of the dirac equation is considered after checking that it recovers the usual discrete schrodinger equation in the nonrelativistic limit it is found that for two valued bernoulli potentials the zero mass case presents the absence of dynamical localization for some specific values of the energy albeit it has no continuous spectrum for the other energy values again excluding some very specific ones the bernoulli dirac system is localized independently of the mass

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

constructive proof of localization in the anderson tight binding model we prove that for large disorder or near the band tails the spectrum of the anderson tight binding hamiltonian with diagonal disorder consists exclusively of discrete eigenvalues the corresponding eigenfunctions are exponentially well localized these results hold in arbitrary dimension and with probability one in one dimension we recover the result that all states are localized for arbitrary energies and arbitrarily small disorder our techniques extend to other physical systems which exhibit localization phenomena such as infinite systems of coupled harmonic oscillators or random schrodinger operators in the continuum

a lagrange multiplier rule with small convex valued subdifferentials for nonsmooth problems of mathematical programming involving equality and nonfunctional constraints it is shown that a lagrange multiplier rule involving the michel penot subdifferentials is valid for the problem minimizef0 x subject tofi x 0 i 1 m fi x 0 i m 1 n x q where all functionsf are lipschitz continuous andq is a closed convex set the proof is based on the theory of fans

dynamical lower bounds for 1d dirac operators quantum dynamical lower bounds for continuous and discrete one dimensional dirac operators are established in terms of transfer matrices then such results are applied to various models including the bernoulli dirac one and in contrast to the discrete case critical energies are also found for the continuous dirac case with positive mass

finite volume fractional moment criteria for anderson localization a technically convenient signature of localization exhibited by discrete operators with random potentials is exponential decay of the fractional moments of the green function within the appropriate energy ranges known implications include spectral localization absence of level repulsion strong form of dynamical localization and a related condition which plays a significant role in the quantization of the hall conductance in two dimensional fermi gases we present a family of finite volume criteria which under some mild restrictions on the distribution of the potential cover the regime where the fractional moment decay condition holds the constructive criteria permit to establish this condition at spectral band edges provided there are sufficient lifshitz tail estimates on the density of states they are also used here to conclude that the fractional moment condition and thus the other manifestations of localization are valid throughout the regime covered by the multiscale analysis in the converse direction the analysis rules out fast power law decay of the green functions at mobility edges

a supersymmetric transfer matrix and differentiability of the density of states in the one dimensional anderson model let h v on 2 z where v x 9 xez are d r v s with common probability distribution v let h t e itvdv v and let k e be the integrated density of states it is proven i if h is n times differentiable with h j t 0 l t for some 0 7 0 l n then k e is a cn function in particular if v has compact support and h t 0 l h 1 with 0 then k e is c this allows v to be singular continuous ii lfh t 0 e a tl for some 0 then k e is analytic in a strip about the real axis the proof uses the supersymmetric replica trick to rewrite the averaged green s function as a two point function of a one dimensional supersymmetric field theory which is studied by the transfer matrix method

new approximation method in the proof of the maximum principle for nonsmooth optimal control problems with state constraints abstract traditional proofs of the pontryagin maximum principle pmp require the continuous differentiability of the dynamics with respect to the state variable on a neighborhood of the minimizing state trajectory when arbitrary values of control variable are inserted into the dynamic equations sussmann has drawn attention to the fact that the pmp remains valid when the dynamics are differentiable with respect to the state variable merely when the minimizing control is inserted into the dynamic equations this weakening of earlier hypotheses has been referred to as the lojasiewicz refinement arutyunov and vinter showed that these extensions of early versions of the pmp can be simply proved by finite dimensional approximations application of a lagrange multiplier rule in finite dimensions and passage to the limit this paper generalizes the finite dimensional approximation technique to a problem with state constraints where the use of needle variations of the optimal control had not been successful moreover the cost function and endpoint constraints are not assumed to be differentiable but merely locally lipschitz continuous the maximum principle is expressed in terms of michel penot subdifferential

anderson localization for one and quasi one dimensional systems we prove almost sure exponential localization of all the eigenfunctions and nondegeneracy of the spectrum for random discrete schrodinger operators on one and quasi one dimensional lattices this paper provides a much simpler proof of these results than previous approaches and extends to a much wider class of systems we remark in particular that the singular continuous spectrum observed in some quasiperiodic systems disappears under arbitrarily small local perturbations of the potential our results allow us to prove that e g for strong disorder the smallest positive lyapunov exponent of some products of random matrices does not vanish as the size of the matrices increases to infinity

on absence of diffusion near the bottom of the spectrum for a random schrodinger operator on l 2 bf r nu we consider a random schrodinger operator onl2 v of the form h omega delta v omega v omega x sigma chi c i x q i omega ci being a covering of v with unit cubes around the sites of v and qi i i d random variables with values in 0 1 we assume that theqi s are continuously distributed with bounded densityf q and that 0 p q0 1 2 1 then we show that an ergodic mean of the quantity dx x 2 exp ith x 2 t 1 vanishes provided ge h where is well localized around the origin andge is a positivec function with support in 0 e e e f our estimate ofe f is such that the set x v v x e f may contain with probability one an infinite cluster of cubes ci which are nearest neighbours the proof is based on the technique introduced by frohlich and spencer for the analysis of the anderson model

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

second order optimality conditions for singular extremals in optimal control problems with equality endpoint constraints abstract in this paper we use the method of finite dimensional approximations to derive the second order necessary optimality conditions for singular pontryagin local minimizers in optimal control problems with equality endpoint constraints

remark on the absence of absolutely continuous spectrum for d dimensional schrodinger operators with random potential for large disorder or low energy we show that there is no absolutely continuous part in the spectrum of the anderson tight binding model for large disorder or low energy the proof is based on the exponential decay of the green s function proved by frohlich and spencer the extension of this result to the continuous case is also discussed

sparse 1d discrete dirac operators i quantum transport abstract some dynamical lower bounds for one dimensional discrete dirac operators with different classes of sparse potentials are presented and the particular role of the particle mass is emphasized

sparse one dimensional discrete dirac operators ii spectral properties we study spectral properties of some discrete dirac operators with nonzero potential only at some sparse and suitably randomly distributed positions as observed in the corresponding schrodinger operators we determine the hausdorff dimension of its spectral measure and identify a sharp spectral transition from point to singular continuous