minimal dynamical systems on the product of the cantor set and the circle we prove that a crossed product algebra arising from a minimal dynamical system on the product of the cantor set and the circle has real rank zero if and only if the set of invariant measures of the system come from the associated cantor minimal system in the case that cocycles take values in the rotation group it is also shown that this condition implies tracial rank zero and in particular the crossed product algebra is isomorphic to a unital simple at algebra of real rank zero under the same assumption we show that two systems are approximately k conjugate if and only if there exists a sequence of isomorphisms between two associated crossed products which approximately maps open image in new window

a kinetic theory for quantum transport in aperiodic media transport theory is reinvestigated in a one particle model including dissipation mechanisms through a phenomenological time dependent part using the description of an aperiodic crystal in terms of noncommutative geometry it is then possible to rigorously derive kubo formulas for the electric conductivity within the framework of the relaxation time approximation a fractal analysis applied to the spectral measures entering in the kubo formula allows us to examine anomalous transport owing to quantum interferences this leads to anomalies in drude s formula

a numerical study of the self similar solutions of the schrodinger map we present a numerical study of the self similar solutions of the localized induction approximation of a vortex filament these self similar solutions which are explicitly known constitute a one parameter family and develop a singularity in finite time we study a number of boundary conditions that allow us to reproduce the mechanism of singularity formation some related questions are also considered

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

extremal eigenvalues and eigenvectors of deformed wigner matrices we consider random matrices of the form h w lambda v lambda in mathbb r where w is a real symmetric or complex hermitian wigner matrix of size n and v is a real bounded diagonal random matrix of size n with i i d entries that are independent of w we assume subexponential decay of the distribution of the matrix entries of w and we choose lambda sim 1 so that the eigenvalues of w and lambda v are typically of the same order further we assume that the density of the entries of v is supported on a single interval and is convex near the edges of its support in this paper we prove that there is lambda in mathbb r such that the largest eigenvalues of h are in the limit of large n determined by the order statistics of v for lambda lambda in particular the largest eigenvalue of h has a weibull distribution in the limit n rightarrow infty if lambda lambda moreover for n sufficiently large we show that the eigenvectors associated to the largest eigenvalues are partially localized for lambda lambda while they are completely delocalized for lambda lambda similar results hold for the lowest eigenvalues

self similar solutions for the 1 d schrodinger map on the hyperbolic plane in this paper we will study self similar solutions for x t x x ss the equivalent in the minkowski 3 space mathbb r 2 1 to the localized induction approximation flow trying to adapt some results given by gutierrez rivas and vega we will show the existence of a one parameter family of smooth solutions developing a corner in finite time the main difference with respect to the euclidean case studied by those authors will be the proof of the boundedness of t e 1 and e 2 the equivalents of t b and n in mathbb r 2 1

z stability of crossed products by strongly outer actions ii we consider a crossed product of a unital simple separable nuclear stably finite z stable c algebra a by a strongly outer cocycle action of a discrete countable amenable group under the assumption that a has finitely many extremal tracial states and is elementary amenable we show that the twisted crossed product c algebra is z stable as an application we also prove that all strongly outer cocycle actions of the klein bottle group on z are cocycle conjugate to each other this is the first classification result for actions of non abelian infinite groups on stably finite c algebras

a characterization of the anderson metal insulator transport transition we investigate the anderson metal insulator transition for random schrodinger operators we define the strong insulator region to be the part of the spectrum where the random operator exhibits strong dynamical localization in the hilbert schmidt norm we introduce a local transport exponent e and set the weak metallic transport region to be the part of the spectrum with nontrivial transport i e e 0 we prove that these insulator and metallic regions are complementary sets in the spectrum of the random operator and that the local transport exponent e provides a characterization of the metal insulator transport transition moreover we show that if there is such a transition then e has to be discontinuous at a transport mobility edge more precisely we show that if the transport is nontrivial then e 1 2d where d is the space dimension these results follow from a proof that slow transport of quantum waves in random media implies the starting hypothesis for the authors bootstrap multiscale analysis we also conclude that the strong insulator region coincides with the part of the spectrum where we can perform a bootstrap multiscale analysis proving that the multiscale analysis is valid all the way up to a transport mobility edge

eigenvector localization for random band matrices with power law band width it is shown that certain ensembles of random matrices with entries that vanish outside a band around the diagonal satisfy a localization condition on the resolvent which guarantees that eigenvectors have strong overlap with a vanishing fraction of standard basis vectors provided the band width w raised to a power remains smaller than the matrix size n for a gaussian band ensemble with matrix elements given by i i d centered gaussians within a band of width w the estimate 8 holds

a new proof of localization in the anderson tight binding model we give a new proof of exponential localization in the anderson tight binding model which uses many ideas of the frohlich martinelli scoppola and spencer proof but is technically simpler particularly the probabilistic estimates

localization and universality of poisson statistics for the multidimensional anderson model at weak disorder we prove anderson localization with the mean field lyapunov exponent and poisson statistics for eigenvalue spacing for the multi dimensional anderson model at weak disorder these results are obtained by developing the supersymmetric formalism initiated in w1 see also sjw rid

classification of homomorphisms and dynamical systems let a be a unital simple c algebra with tracial rank zero and let x be a compact metric space suppose that h 1 h 2 c x a are two unital monomorphisms we show that h 1 and h 2 are approximately unitarily equivalent if and only if h 1 h 2 in kl c x a and o h 1 f o h 2 f for every f c x and every trace r of a inspired by a theorem of tomiyama we introduce a notion of approximate conjugacy for minimal dynamical systems let x be a compact metric space and let x x be two minimal homeomorphisms using the above mentioned result we show that two dynamical systems are approximately conjugate in that sense if and only if a k theoretical condition is satisfied in the case that x is the cantor set this notion coincides with the strong orbit equivalence of giordano putnam and skau and the k theoretical condition is equivalent to saying that the associate crossed product c algebras are isomorphic another application of the above mentioned result is given for c dynamical systems related to a problem of kishimoto let a be a unital simple ah algebra with no dimension growth and with real rank zero and let a aut a we prove that if r fixes a large subgroup of k 0 a and has the tracial rokhlin property then ax z is again a unital simple ah algebra with no dimension growth and with real rank zero

delocalization for a class of random block band matrices we consider n times n hermitian random matrices h consisting of blocks of size m ge n 6 7 the matrix elements are i i d within the blocks close to a gaussian in the four moment matching sense but their distribution varies from block to block to form a block band structure with an essential band width m we show that the entries of the green s function g z h z 1 satisfy the local semicircle law with spectral parameter z e mathbf i eta down to the real axis for any eta gg n 1 using a combination of the supersymmetry method inspired by shcherbina j stat phys 155 3 466 499 2014 and the green s function comparison strategy previous estimates were valid only for eta gg m 1 the new estimate also implies that the eigenvectors in the middle of the spectrum are fully delocalized

the large connectivity limit of the anderson model on tree graphs we consider the anderson localization problem on the infinite regular tree within the localized phase we derive a rigorous lower bound on the free energy function recently introduced by aizenman and warzel using a finite volume regularization we also derive an upper bound on this free energy function this yields upper and lower bounds on the critical disorder such that all states at a given energy become localized these bounds are particularly useful in the large connectivity limit where they match confirming the early predictions of abou chacra anderson and thouless

the rohlin property for shifts on uhf algebras and automorphisms of cuntz algebras abstract we show that the shift on the uhf algebra m n has the rohlin property for any odd n hence by brotteli et al the crossed product of a uhf algebra by a shift ergodic theory dynam systems 13 1993 615 626 for any n 2 3 and that any automorphism of the cuntz algebra o n n m is outer for any m 0

singular continuous spectrum is generic in a variety of contexts we prove that singular continuous spectrum is generic in the sense that for certain natural complete metric spaces of operators those with singular spectrum are a dense g delta

stability of the self similar dynamics of a vortex filament in this paper we continue our investigation of self similar solutions of the vortex filament equation also known as the binormal flow or the localized induction equation our main result is the stability of the self similar dynamics of small perturbations of a given self similar solution the proof relies on finding precise asymptotics in space and time for the tangent and the normal vectors of the perturbations a main ingredient in the proof is the control of the evolution of weighted norms for a cubic one dimensional schrodinger equation connected to the binormal flow by hasimoto s transform

non commutative shifts and crossed products abstract when a is a unital simple af c algebra and has a unique tracial state it is shown that the crossed product of the two sided infinite tensor product z a by the shift is a tracially af c algebra a similar result is given to the crossed product of a certain non unital two sided infinite tensor product by the shift applying a far reaching classification result of such c algebras by h lin we obtain an example of a one parameter automorphism group on some af c algebra which is not approximately inner a counter example to the af version of the so called powers sakai conjecture 23

chiral vector bundles this paper focuses on the study of a new category of vector bundles the objects of this category called chiral vector bundles are pairs given by a complex vector bundle along with one of its automorphisms we provide a classification for the homotopy equivalence classes of these objects based on the construction of a suitable classifying space the computation of the cohomology of the latter allows us to introduce a proper set of characteristic cohomology classes some of those just reproduce the ordinary chern classes but there are also new odd dimensional classes which take care of the extra topological information introduced by the chiral structure chiral vector bundles provide a geometric model for topological quantum systems in class aiii namely for systems endowed with a pseudo symmetry of chiral type the classification of the chiral vector bundles over spheres and tori explicitly computable up to dimension 4 recovers the commonly accepted classification for topological insulators of class aiii which is usually based on the k group k 1 however our classification turns out to be even richer since it takes care also for possible non trivial chern classes

localization for random and quasiperiodic potentials a survey is made of some recent mathematical results and techniques for schrodinger operators with random and quasiperiodic potentials a new proof of localization for random potentials established in collaboration with h von dreifus is sketched

localization in general one dimensional random systems i jacobi matrices we consider random discrete schrodinger operators in a strip with a potentialv n n a label in and a finite label across the strip andv an ergodic process we prove thath0 v has only point spectrum with probability one under two assumptions 1 theconditional distribution of v n n 0 1 all conditioned on v n 0 1 all has an absolutely continuous component with positive probability 2 for a e e no lyaponov exponent is zero

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

singular continuous spectrum under rank one perturbations and localization for random hamiltonians we consider a selfadjoint operator a and a selfadjoint rank one projection p onto a vector which is cyclic for a in terms of the spectral measure d a we give necessary and sufficient conditions for a p to have empty singular continuous spectrum or to have only point spectrum for a e we apply these results to questions of localization in the one and multi dimensional anderson models

classification of certain infinite simple c algebras the class of c algebras that arise as the crossed product of a stable simple af algebra with a z action determined by an automorphism which maps a projection in the algebra onto a proper subprojection is proved to consist of simple purely infinite c algebras and a specific subclass of it is proved to be classified by k theory this subclass is large enough to exhaust all possible k groups if g0 and g1 are countable abelian groups with g1 torsion free as it must be then there is a c algebras a in the classified subclass with k0 a g0 and k 1 a g1 the subclass contains the cuntz algebras o with n even and the cuntz krieger algebras o a with k0 o a of finite old order and it is closed under forming inductive limits the c algebras in the classified subclass can be viewed as classifiable models in a strong sense of general simple purely infinite c algebras with the same k theory

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

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

stability for integer actions on uhf c algebras on montre qu il est possibile d obtenir une version norme continue de la solution de connes a la 1 cohomologie qui est adaptable a la situation des c algebres on exhibe diverses actions modeles stables

localization for random schrodinger operators with correlated potentials we prove localization at high disorder or low energy for lattice schrodinger operators with random potentials whose values at different lattice sites are correlated over large distances the class of admissible random potentials for our multiscale analysis includes potentials with a stationary gaussian distribution whose covariance functionc x y decays as x y where 0 can be arbitrarily small and potentials whose probability distribution is a completely analytical gibbs measure the result for gaussian potentials depends on a multivariable form of nelson s best possible hypercontractive estimate

mathbb z actions on ah algebras and mathbb z 2 actions on af algebras we consider mathbb z actions single automorphisms on a unital simple ah algebra with real rank zero and slow dimension growth and show that the uniform outerness implies the rohlin property under some technical assumptions moreover two mathbb z actions with the rohlin property on such a c algebra are shown to be cocycle conjugate if they are asymptotically unitarily equivalent we also prove that locally approximately inner and uniformly outer mathbb z 2 actions on a unital simple af algebra with a unique trace have the rohlin property and classify them up to cocycle conjugacy employing the orderext group as classification invariants

outer automorphisms and reduced crossed products of simple c algebras every outer automorphism of a separable simplec algebra is shown to have a pure state which is mapped into an inequivalent state under this automorphism the reduced crossed product of a simplec algebra by a discrete group of outer automorphisms is shown to be simple

numerical methods for the landau lifshitz equation we discuss numerical schemes for various forms of the landau lifshitz equation a new simple projection method is introduced and is shown to be unconditionally stable the advantages over other schemes are also demonstrated numerically

delocalization at small energy for heavy tailed random matrices we prove that the eigenvectors associated to small enough eigenvalues of an heavy tailed symmetric random matrix are delocalized with probability tending to one as the size of the matrix grows to infinity the delocalization is measured thanks to a simple criterion related to the inverse participation ratio which computes an average ratio of l 4 and l 2 norms of vectors in contrast as a consequence of a previous result for random matrices with sufficiently heavy tails the eigenvectors associated to large enough eigenvalues are localized according to the same criterion the proof is based on a new analysis of the fixed point equation satisfied asymptotically by the law of a diagonal entry of the resolvent of this matrix

unbounded derivations in at algebras letabe a simple unital at algebra of real rank zero such that it has a unique tracial state andk1 a is neither 0 norz for each hom k1 a r with dense range inrwe construct a closed derivation inawhich generates a one parameter automorphism group ofasuch that u u 2 i u for any unitaryu d furthermore we construct such an with the rohlin property which is defined in kishimoto comm math phys 179 1996 599 622 in this case the crossed producta ris a simple at algebra of real rank zero as an application we obtain that for such ac algebraathe kernel of the natural homomorphism of the groupinn a of approximately inner automorphisms intoext k1 a k0 a ext k0 a k1 a is the group hinn a of automorphisms homotopic to inner automorphisms combining with the result of kishimoto and kumjian trans amer math soc to appear inn a hinn a is isomorphic to the above direct sum as another application of the construction of derivations we show that ifais ac algebra of the above type and hinn a has the rohlin property and comes from hom k1 a r with dense range as in kishimoto and kumjian preprint then the crossed producta zis again of the same type in particulara zis an at algebra the other properties are known from kishimoto j operator theory40 1998

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

spectral properties of random schrodinger operators with unbounded potentials we investigate spectral properties of random schrodinger operators h n 1 n acting onl 2 z d where n are independent random variables uniformly distributed on 0 1

trace class perturbations and the absence of absolutely continuous spectra we show that various hamiltonians and jacobi matrices have no absolutely continuous spectrum by showing that under a trace class perturbation they become a direct sum of finite matrices

absolutely continuous spectrum for the anderson model on a tree a geometric proof of klein s theorem we give a new proof of a version of klein s theorem on the existence of absolutely continuous spectrum for the anderson model on the bethe lattice at weak disorder

a central limit theorem for products of random matrices and goe statistics for the anderson model on long boxes we consider products of random matrices that are small independent identically distributed perturbations of a fixed matrix mathcal t 0 focusing on the eigenvalues of mathcal t 0 of a particular size we obtain a limit to a sde in a critical scaling previous results required mathcal t 0 to be a conjugated unitary matrix so it could not have eigenvalues of different modulus from the result we can also obtain a limit sde for the markov process given by the action of the random products on the flag manifold applying the result to random schrodinger operators we can improve some results by valko and virag showing goe statistics for the rescaled eigenvalue process of a sequence of anderson models on long boxes in particular we solve a problem posed in their work

crossed products by minimal homeomorphisms let x be an infinite compact metric space with finite covering dimension and let h x x be a minimal homeomorphism we show that the associated crossed product c algebra a 1 4 c z x h has tracial rank zero whenever the image of k0 a in a t a is dense as a consequence we show that these crossed product c algebras are in fact simple ah algebras with real rank zero when x is connected and h is further assumed to be uniquely ergodic then the above happens if and only if the rotation number associated to h has irrational values by applying the classification result for nuclear simple c algebras with tracial rank zero we show that two such dynamical systems have isomorphic crossed products if and only if they have isomorphic scaled ordered k theory

non commutative odd chern numbers and topological phases of disordered chiral systems abstract an index theorem for higher chern characters of odd fredholm modules over crossed product algebras is proved together with a local formula for the associated cyclic cocycle the result generalizes the classic noether gohberg krein index theorem which in its simplest form states that the winding number of a complex valued function over the circle is equal to the index of the associated toeplitz operator when applied to the non commutative brillouin zone this generalization allows to define topological invariants for all condensed matter phases from the chiral unitary or aiii symmetry class in the presence of strong disorder and magnetic fields whenever the fermi level lies in a region of anderson localized spectrum

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

quantum harmonic oscillator systems with disorder we study many body properties of quantum harmonic oscillator lattices with disorder a sufficient condition for dynamical localization expressed as a zero velocity lieb robinson bound is formulated in terms of the decay of the eigenfunction correlators for an effective one particle hamiltonian we show how state of the art techniques for proving anderson localization can be used to prove that these properties hold in a number of standard models we also derive bounds on the static and dynamic correlation functions at both zero and positive temperature in terms of one particle eigenfunction correlators in particular we show that static correlations decay exponentially fast if the corresponding effective one particle hamiltonian exhibits localization at low energies regardless of whether there is a gap in the spectrum above the ground state or not our results apply to finite as well as to infinite oscillator systems the eigenfunction correlators that appear are more general than those previously studied in the literature in particular we must allow for functions of the hamiltonian that have a singularity at the bottom of the spectrum we prove exponential bounds for such correlators for some of the standard models

upper bound on the energy gap in terms of the localization in tight binding systems we present an upper bound on the energy gap above the ground energy in terms of the variance of the particle position in one particle tight binding systems our inequality gives a complementary relationship between the energy gap and the localization of the particle that is the energy gap narrows as the ground state spreads spatially while the energy gap can broaden as the ground state becomes localized we first consider the ground state in one dimensional tight binding systems and relate the energy gap to the variance of the particle position in the ground state we then extend the result to higher dimensional systems we compare our upper bound with the exact energy gap as calculated numerically in one and two dimensions this comparison confirms that our upper bound places a tight restriction on the energy gap

on mott s formula for the ac conductivity in the anderson model we study the ac conductivity in linear response theory in the general framework of ergodic magnetic schrodinger operators for the anderson model if the fermi energy lies in the localization regime we prove that the ac conductivity is bounded from above by c 2 log 1 d 2 at small frequencies this is to be compared to mott s formula which predicts the leading term to be c 2 log 1 d 1

the rokhlin property and the tracial topological rank abstract let a be a unital separable simple c algebra with tr a 1 and be an automorphism we show that if satisfies the tracially cyclic rokhlin property then tr a z 1 we also show that whenever a has a unique tracial state and m is uniformly outer for each m 0 and r is approximately inner for some r 0 satisfies the tracial cyclic rokhlin property by applying the classification theory of nuclear c algebras we use the above result to prove a conjecture of kishimoto if a is a unital simple a t algebra of real rank zero and aut a which is approximately inner and if satisfies some rokhlin property then the crossed product a z is again an a t algebra of real rank zero as a by product we find that one can construct a large class of simple c algebras with tracial rank one and zero from crossed products

removing the stiffness from interfacial flows with surface tension a new formulation and new methods are presented for computing the motion of fluid interfaces with surface tension in two dimensional irrotational and incompressible fluids through the laplace young condition at the interface surface tension introduces high order terms both nonlinear and nonlocal into the dynamics this leads to severe stability constraints for explicit time integration methods and makes the application of implicit methods difficult this new formulation has all the nice properties for time integration methods that are associated with having a linear constant coefficient highest order term that is using this formulation we give implicit time integration methods that have no high order time step stability constraint associated with surface tension and are explicit in fourier space the approach is based on a boundary integral formulation and applies more generally even to problems beyond the fluid mechanical context here they are applied to computing with high resolution the motion of interfaces in hele shaw flows and the motion of free surfaces in inviscid flows governed by the euler equations one hele shaw computation shows the behavior of an expanding gas bubble over long time as the interface undergoes successive tip splittings and finger competition a second computation shows the formation of a very ramified interface through the interaction of surface tension with an unstable density stratification in euler flows the computation of a vortex sheet shows its roll up through the kelvin helmholtz instability this motion culminates in the late time self intersection of the interface creating trapped bubbles of fluid this is we believe a type of singularity formation previously unobserved for such flows in 2d finally computations of falling plumes in an unstably stratified boussinesq fluid show a very similar behavior

simplicity of eigenvalues in the anderson model we give a transparent and intuitive proof that all eigenvalues of the anderson model in the region of localization are simple

integrated density of states for random metrics on manifolds we study ergodic random schrodinger operators on a covering manifold where the randomness enters both via the potential and the metric we prove measurability of the random operators almost sure constancy of their spectral properties the existence of a self averaging integrated density of states and a pastur ubin type trace formula

schrodinger operators in the twentieth century this paper reviews the past fifty years of work on spectral theory and related issues in nonrelativistic quantum mechanics

diffusion in the mean for an ergodic schrodinger equation perturbed by a fluctuating potential diffusive scaling of position moments and a central limit theorem are obtained for the mean position of a quantum particle hopping on a cubic lattice and subject to a random potential consisting of a large static part and a small part that fluctuates stochastically in time if the static random potential is strong enough to induce complete localization in the absence of time dependent noise then the diffusion constant is shown to go to zero proportional to the square of the strength of the time dependent part