quantum distribution functions for radial observables for quantum systems with two dimensional configuration space we construct a physical radial momentum observable rescaling the radius we find that the dilation degrees of freedom form a weyl algebra with this we construct a radial wigner quasiprobability distribution function

right limits and reflectionless measures for cmv matrices we study cmv matrices by focusing on their right limit sets we prove a cmv version of a recent result of remling dealing with the implications of the existence of absolutely continuous spectrum and we study some of its consequences we further demonstrate the usefulness of right limits in the study of weak asymptotic convergence of spectral measures and ratio asymptotics for orthogonal polynomials by extending and refining earlier results of khrushchev to demonstrate the analogy with the jacobi case we recover corresponding previous results of simon using the same approach

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

hydrogen atom in phase space the wigner representation the hydrogen atom is a fundamental exactly soluble system for which the wigner function being a quantum analogue of the joint probability distribution of position and momentum is unknown in this paper we present an effective method of calculating the wigner function for all bound states of the nonrelativistic hydrogen atom the formal similarity between the eigenfunctions of the nonrelativistic hydrogen atom in the momentum representation and the klein gordon propagator has allowed the calculation of the wigner function for an arbitrary bound state of the hydrogen atom using a simple atomic integral as a generator these wigner functions for some low lying states are depicted and discussed

wigner function for the radial equation of the coulomb problem in langer space in this paper a theoretical study of the coulomb problem in quantum phase space is reported the langer coordinate transformation is used to map the coulomb problem into a one dimensional morse oscillator as a result the wigner distribution functions for the morse oscillator are obtained the form of these functions is presented for a few principal quantum number n and the angular momentum quantum number l the results obtained correspond to the solution of the coulomb problem in the spherical coordinate system

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

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

an exactly solvable model of a multidimensional incommensurate structure the paper considers the class of schrodinger multidimensional discrete operators with quasi periodic unbounded potential for which essentially complete spectral analysis may be carried out in the case of sufficiently high incommensurability of almost periods the spectrum of such operators is found to be pure point and simple the eigenfunctions exponentially localized and the low frequency conductivity exponentially small in the one dimensional case for any incommensurability the spectrum does not contain the absolutely continuous component while for small incommensurability the spectrum is singular continuous

approach to the extended states conjecture we develop a rather explicit approach concerning the extended states conjecture for the discrete random schrodinger operator or more generally for the so called anderson type hamiltonian our work is based on deep mathematical results by jaksic last duke math j 133 1 185 204 2006 concretely we suggest two new directions of research we provide a formula which may lead the way to a rigorous proof of the conjecture and an implementation of the proposed approach which yields numerical evidence in favor of the conjecture being true for the discrete random schrodinger operator in dimension two not being based on scaling theory this method eliminates problems due to boundary conditions common to previous numerical methods in the field at the same time as with any numerical experiment one cannot exclude finite size effects with complete certainty we numerically track the bulk distribution here the distribution of where we most likely find an electron of a wave packet initially located at the origin after iterative application of the discrete random schrodinger operator

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

dynamical delocalization in random landau hamiltonians we prove the existence of dynamical delocalization for random landau hamiltonians near each landau level since typically there is dynamical localization at the edges of each disordered broadened landau band this implies the existence of at least one dynamical mobility edge at each landau band namely a boundary point between the localization and delocalization regimes which we prove to converge to the corresponding landau level as either the magnetic field goes to infinity or the disorder goes to zero in this article we prove the existence of dynamical delocalization for random landau hamiltonians near each landau level more precisely we prove that for these two dimensional hamiltonians there exists at least one energy e near each landau level such that e 1 where e the local transport exponent introduced in gk5 is a measure of the rate of transport in wave packets with spectral support near e since typically there is dynamical localization at the edges of each disordered broadened landau band this implies the existence of at least one dynamical mobility edge at each landau band namely a boundary point between the localization and delocalization regimes which we prove to converge to the corresponding landau level as either the magnetic field goes to infinity or the disorder goes to zero random landau hamiltonians are the subject of intensive study due to their links with the integer quantum hall e ect kli for which von klitzing received the 1985 nobel prize in physics they describe an electron moving in a very thin flat conductor with impurities under the influence of a constant magnetic field perpendicular to the plane of the conductor and play an important role in the understanding of the quantum hall e ect l aoa t h nt ku be avss bees laughlin s argument l as pointed out by halperin h uses the assumption that under weak disorder and strong magnetic field the energy spectrum consists of bands of extended states separated

random dirac operators with time reversal symmetry quasi one dimensional stochastic dirac operators with an odd number of channels time reversal symmetry but otherwise efficiently coupled randomness are shown to have one conducting channel and absolutely continuous spectrum of multiplicity two this follows by adapting the criteria of guivarch raugi and goldsheid margulis to the analysis of random products of matrices in the group so 2l and then a version of kotani theory for these operators absence of singular spectrum can be shown by adapting an argument of jaksic last if the potential contains random dirac peaks with absolutely continuous distribution

dynamical localization for discrete and continuous random schrodinger operators we show for a large class of random schrodinger operators h on and on that dynamical localization holds i e that with probability one for a suitable energy interval i and for q a positive real here is a function of sufficiently rapid decrease and p i h is the spectral projector of h corresponding to the interval i the result is obtained through the control of the decay of the eigenfunctions of h and covers in the discrete case the anderson tight binding model with bernoulli potential dimension 1 or singular potential 1 and in the continuous case anderson as well as random landau hamiltonians

simple strong glass forming models mean field solution with activation we introduce simple models inspired by previous models for froths and covalent glasses with trivial equilibrium properties but dynamical behaviour characteristic of strong glass forming systems these models are also a generalization of backgammon or urn models to a non constant number of particles where entropic barriers are replaced by energy barriers allowing for the existence of activated processes we formulate a mean field version of the models which keeps most of the features of the finite dimensional ones and solve analytically the out of equilibrium dynamics in the low temperature regime where activation plays an essential role

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

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

boosted simon wolff spectral criterion and resonant delocalization discussed here are criteria for the existence of continuous components in the spectra of operators with random potential first the essential condition for the simon wolff criterion is shown to be measurable at infinity by implication for the i i d case and more generally potentials with the k property the criterion is boosted by a zero one law the boosted criterion combined with tunneling estimates is then applied for sufficiency conditions for the presence of continuous spectrum for random schrodinger operators the general proof strategy that this yields is modeled on the resonant delocalization arguments by which continuous spectrum in the presence of disorder was previously established for random operators on tree graphs in another application of the simon wolff rank one analysis we prove the almost sure simplicity of the pure point spectrum for operators with random potentials of conditionally continuous distribution 2015 wiley periodicals inc

metal insulator transition for the almost mathieu operator we prove that for diophantine and almost every the almost mathieu operator h n n 1 ni 1 cos 2 n n exhibits localization for 2 and purely absolutely continuous spectrum for 2

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

wigner function statistics in classically chaotic systems we have studied statistical properties of the values of the wigner function w x of 1d quantum maps on compact 2d phase space of finite area v for this purpose we have defined a wigner function probability distribution p w 1 v w w x dx which has by definition fixed first and second moments in particular we concentrate on relaxation of time evolving quantum states in terms of w x starting from a coherent state we have shown that for a classically chaotic quantum counterpart the distribution p w in the semiclassical limit becomes a gaussian distribution that is fully determined by the first two moments numerical simulations have been performed for the quantum sawtooth map and the quantized kicked top in a quantum system with hilbert space dimension n 1 the transition of p w to a gaussian distribution was observed at times t log n in addition it has been shown that the statistics of wigner functions of propagator eigenstates is gaussian as well in the classically fully chaotic regime we have also studied the structure of the nodal cells of the wigner function in particular the distribution of intersection points between the zero manifold and arbitrary straight lines

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

multiplicity bound of singular spectrum for higher rank anderson models abstract in this work we prove a bound on the multiplicity of the singular spectrum for a certain class of anderson hamiltonians the operator in consideration is the form h n z d n p n on the hilbert space l 2 z d where is discrete laplacian p n are projection onto l 2 x z d n i l i x i n i 1 l i for some l 1 l d n and n n are i i d real bounded random variables following an absolutely continuous distribution we prove that the multiplicity of the singular spectrum is bounded above by 2 d d independent of l i i 1 d when l i 1 2 n 3 n for all i and g c d l i 1 l j 1 1 for i j the singular spectrum is also simple

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

asymptotics of radial wave equations the langer modification is an improvement in the wkb analysis of the radial schrodinger equation we derive a generalization of the langer modification to any radial operator for differential operators we write the modified classical symbols explicitly and show that the wkb wavefunctions with the modification have the exact limiting behavior for small radius unlike in the schrodinger case generally the modified radial analysis is not equivalent to the wkb analysis of the full problem before reduction by the spherical symmetry

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

almost periodic schrodinger operators iii the absolutely continuous spectrum in one dimension we studyh d2 dx2 v x withv x limit periodic e g v x ancos x 2n with an we prove that for a genericv and for generican in the explicit example h is a cantor nowhere dense perfect set for a dense set the spectrum is both cantor and purely absolutely continuous and therefore purely recurrent absolutely continuous

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