a bloch decomposition based stochastic galerkin method for quantum dynamics with a random external potential in this paper we consider the numerical solution of the one dimensional schrodinger equation with a periodic lattice potential and a random external potential this is an important model in solid state physics where the randomness results from complicated phenomena that are not exactly known here we generalize the bloch decomposition based time splitting pseudospectral method to the stochastic setting using the generalized polynomial chaos with a galerkin procedure so that the main effects of dispersion and periodic potential are still computed together we prove that our method is unconditionally stable and numerical examples show that it has other nice properties and is more efficient than the traditional method finally we give some numerical evidence for the well known phenomenon of anderson localization

quasi valuations extending a valuation abstract suppose f is a field with valuation v and valuation ring o v e is a finite field extension and w is a quasi valuation on e extending v we study quasi valuations on e that extend v in particular their corresponding rings and their prime spectra we prove that these ring extensions satisfy inc incomparability lo lying over and gd going down over o v in particular they have the same krull dimension we also prove that every such quasi valuation is dominated by some valuation extending v under the assumption that the value monoid of the quasi valuation is a group we prove that these ring extensions satisfy gu going up over o v and a bound on the size of the prime spectrum is given in addition a one to one correspondence is obtained between exponential quasi valuations and integrally closed quasi valuation rings given r an algebra over o v we construct a quasi valuation on r we also construct a quasi valuation on r o v f which helps us prove our main theorem the main theorem states that if r e satisfies r f o v and e is the field of fractions of r then r and v induce a quasi valuation w on e such that r o w and w extends v thus r satisfies the properties of a quasi valuation ring

maximal symmetry group of the hamilton jacobi equation relativistic particle in flat spacetime lie s extended group method has been used to obtain the maximal symmetry group of the hamilton jacobi equation for a relativistic particle moving in flat spacetime for a massive particle it is the 21 parameter inhomogeneous pseudo orthogonal group io 4 1 of signature 4 1 for a zero mass particle like a photon or a neutrino this group is an infinite parameter lie group with an infinite parameter invariant subgroup such that the factor group is isomorphic to the inhomogeneous pseudo orthogonal group io 3 1 of signature 3 1

regularity of the invariant measure and of the density of states in the one dimensional anderson model abstract let h 1 2 v on l2 z where v x x z are i i d r v s with common probability distribution let h t e itvd v let n e be the integrated density of states and let e be the invariant measure at energy e it is proven 1 1 let h be d times differentiable on 0 with bounded derivatives such that h j t 0 as t for j 0 1 d then i n e is a cd function ii if d 3 e is absolutely continuous for all e r and d e dx is of class c d 1 2 1 jointly in e and x 2 2 let h be differentiable on 0 with h j t o e b t for j 0 1 and some b 0 then d e dx is jointly analytic in e and x

hamiltonians on discrete structures jumps of the integrated density of states and uniform convergence we study equivariant families of discrete hamiltonians on amenable geometries and their integrated density of states ids we prove that the eigenspace of a fixed energy is spanned by eigenfunctions with compact support the size of a jump of the ids is consequently given by the equivariant dimension of the subspace spanned by such eigenfunctions from this we deduce uniform convergence w r t the spectral parameter of the finite volume approximants of the ids our framework includes quasiperiodic operators on delone sets periodic and random operators on quasi transitive graphs and operators on percolation graphs

fractional moment methods for anderson localization with saw representation the green function contains much information about physical systems mathematically the fractional moment method fmm developed by aizenman and molchanov connects the green function and the transport of electrons in the anderson model recently it has been discovered that the green function on a graph can be represented using self avoiding walks on a graph which allows us to connect localization properties in the system and graph properties we discuss fmm in terms of the self avoiding walks on a general graph the only general condition being that the graph has a uniform bound on the vertex degree

asymptotic analysis of the lyapunov exponent and rotation number of the random oscillator and application we construct asymptotic expansions for the exponential growth rate lyapunov exponent and rotation number of the random oscillator when the noise is large small rapidly varying or slowly varying we then apply our results to problems in the stability of the random oscillator the spectrum of the one dimensional random schr6dinger operator and wave propagation in a one dimensional random medium

functionals of brownian motion localization and metric graphs we review several results related to the problem of a quantum particle in a random environment in an introductory part we recall how several functionals of brownian motion arise in the study of electronic transport in weakly disordered metals weak localization two aspects of the physics of the one dimensional strong localization are reviewed some properties of the scattering by a random potential time delay distribution and a study of the spectrum of a random potential on a bounded domain the extreme value statistics of the eigenvalues then we mention several results concerning the diffusion on graphs and more generally the spectral properties of the schrodinger operator on graphs the interest of spectral determinants as generating functions characterizing the diffusion on graphs is illustrated finally we consider a two dimensional model of a charged particle coupled to the random magnetic field due to magnetic vortices we recall the connection between spectral properties of this model and winding functionals of planar brownian motion

eigenvalue statistics for random schrodinger operators with non rank one perturbations we prove that certain natural random variables associated with the local eigenvalue statistics for generalized lattice anderson models constructed with finite rank perturbations are compound poisson distributed this distribution is characterized by the fact that the levy measure is supported on at most a finite set of positive integers determined by the rank the proof relies on a minami type estimate for finite rank perturbations for anderson type continuum models on mathbb r d we prove a similar result for certain natural random variables associated with the local eigenvalue statistics we prove that the compound poisson distribution associated with these random variables has a levy measure whose support is at most the set of positive integers

spectral flow and level spacing of edge states for quantum hall hamiltonians we consider a non relativistic particle on the surface of a semi infinite cylinder of circumference l submitted to a perpendicular magnetic field of strength b and to the potential of impurities of maximal amplitude w this model is of importance in the context of the integer quantum hall effect in the regime of strong magnetic field or weak disorder b w it is known that there are chiral edge states which are localized within a few magnetic lengths close to and extended along the boundary of the cylinder and whose energy levels lie in the gaps of the bulk system these energy levels have a spectral flow uniform in l as a function of a magnetic flux which threads the cylinder along its axis through a detailed study of this spectral flow we prove that the spacing between two consecutive levels of edge states is bounded below by 2 l 1 with 0 independent of l and of the configuration of impurities this implies that the level repulsion of the chiral edge states is much stronger than that of extended states in the usual anderson model and their statistics cannot obey one of the gaussian ensembles our analysis uses the notion of relative index between two projections and indicates that the level repulsion is connected to topological aspects of quantum hall systems

maximal symmetry groups of quantum relativistic equations the author has used lie s extended group method to obtain the maximal symmetry groups of the dirac equation for finite mass spin 1 2 particles and the weyl equation for zero mass spin 1 2 particles in both cases the maximal symmetry group is an infinite parameter lie group having an invariant subgroup also of an infinite number of generators the corresponding factor group for the dirac equation is an eleven parameter lie group isomorphic to the weyl group in the case of the weyl equation the corresponding factor group is a sixteen parameter lie group containing a proper subgroup isomorphic to the conformal group

a numerical study of sparse random matrices a numerical study is presented for the eigensolution statistics of largen n real and symmetric sparse random matrices as a function of the mean numberp of nonzero elements per row the model shows classical percolation and quantum localization transitions atp c 1 andp q 1 respectively in the rigid limitp n we demonstrate that the averaged density of states follows the wigner semicircle law and the corresponding nearest energy level spacing distribution functionp s obeys the wigner surmise in the very sparse matrix limitp n withp p q a singularity e 1 e is found as e 0 and exponential tails develop in the high e regions but thep s distribution remains consistent with level repulsion the localization properties of the model are examined by studying both the eigenvector amplitude and the density fluctuations the valuep q 1 4 is roughly estimated in agreement with previous studies of the anderson transition in dilute bethe lattices

level statistics in a quasiperiodic system the authors examine the level statistics of harper s equation a simple model of a one dimensional quasiperiodic system through the distribution of the spacings between adjacent eigenvalues each normalised by the average local density of states they present exact results for the distribution for both small and large amplitude alpha of the incommensurate potential showing it has a simple form in each of these limits they show numerically that these distributions are preserved as alpha approaches its critical value 2 from each side and also that at the critical point the distribution of the normalised spacings has a distinct but simple form

the absolute continuity of the integrated density of states for magnetic schrodinger operators with certain unbounded random potentials the object of the present study is the integrated density of states of a quantum particle in multi dimensional euclidean space which is characterized by a schrodinger operator with magnetic field and a random potential which may be unbounded from above and below in case that the magnetic field is constant and the random potential is ergodic and admits a so called one parameter decomposition we prove the absolute continuity of the integrated density of states and provide explicit upper bounds on its derivative the density of states this local lipschitz continuity of the integrated density of states is derived by establishing a wegner estimate for finite volume schrodinger operators which holds for rather general magnetic fields and different boundary conditions examples of random potentials to which the results apply are certain alloy type and gaussian random potentials besides we show a diamagnetic inequality for schrodinger operators with neumann boundary conditions

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

embedding henselian fields into power series abstract every henselian field of residue characteristic 0 admits a truncation closed embedding in a field of generalised power series possibly with a factor set as corollaries we obtain the ax kochen ershov theorem and an extension of mourgues and ressayre s theorem every ordered field which is henselian in its natural valuation has an integer part we also give some results for the mixed and the finite characteristic cases

statistics of quasienergies in chaotic and random systems abstract the statistics of quasienergies are analyzed for periodically driven chaotic systems and found to be similar to those of truly random models these differ from the results that were obtained so far for chaotic systems with time independent hamiltonians the separations of the quasienergies and the 3 statistic are calculated numerically for chaotic as well as for truly random models local statistical measures are introduced in order to investigate the repulsion of quasienergies the results provide further evidence for anderson localization in chaotic systems with hamiltonians that are periodic in time

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

lifshitz tails for periodic plus random potentials w kirsch i and b simon 2 received november 27 1984 we prove that the integrated density of states p 2 for a potential w vper v o has lifshitz tails where vp r is a periodic potential with reflec tion symmetry and v o is a random potential e g of the form v z q o f x i key words lifshitz tails anderson model dirichlet neumann bracketing

liapunov spectra for infinite chains of nonlinear oscillators we argue that the spectrum of liapunov exponents for long chains of nonlinear oscillators at large energy per mode may be well approximated by the liapunov exponents of products of independent random matrices if in addition statistical mechanics applies to the system the elements of these random matrices have a distribution which may be calculated from the potential and the energy alone under a certain isotropy hypothesis which is not always satisfied we argue that the liapunov exponents of these random matrix products can be obtained from the density of states of a typical random matrix this construction uses an integral equation first derived by newman we then derive and discuss a method to compute the spectrum of a typical random matrix putting the pieces together we see that the liapunov spectrum can be computed from the potential between the oscillators

smoothness of the integrated density of states on strips smoothness of the integrated density of states k e of random schrodinger operators i i d and non i i d cases on a discrete strip lattice is investigated it is proven that k e is c if only the potentials on the top surface of the strip have distributions with compactly supported densities in some fractional sobolev space the c result for the case of the anderson model i e all potentials having a distribution with compactly supported density in some sobolev space is also recovered

rank one perturbations and the zeros of paraorthogonal polynomials on the unit circle we prove several results about zeros of paraorthogonal polynomials using the theory of rank one perturbations of unitary operators in particular we obtain new details on the interlacing of zeros for successive popuc

quantum diffusion in the generalized harper equation we study numerically the dynamic and spectral properties of a one dimensional quasi periodic system where site energies are given by with denoting the kth quasiperiodic lattice site when is given by the reciprocal lattice vector with n and m being successive fibonacci numbers the variance of the wavepacket is found to grow quadratically in time regardless of the potential strength v for other values of f there exists a critical value of v beyond which the growth of the wavepacket variance is bounded in particular an anomalous diffusion takes place for corresponding to with generic integers m and n the level spacing distribution is also examined and the corresponding exponent is observed to decrease with v

spectral analysis of selfadjoint jacobi matrices with periodically modulated entries this paper deals with the spectral analysis of a class of selfadjoint unbounded jacobi matrices j with modulated entries the entries have the form of smooth sequences that increase to infinity multiplied by proper periodic sequences for this class criteria for pure absolute continuity of the spectrum or its discreteness and the asymptotics of generalized eigenvectors of j are given some examples illustrating the stability zones of spectral structure are presented

towers of complements to valuation rings and truncation closed embeddings of valued fields abstract we study necessary and sufficient conditions for a valued field k with value group g and residue field k with char k char k to admit a truncation closed embedding in the field of generalized power series k g f with factor set f we show that this is equivalent to the existence of a family tower of complements of k subspaces of k which are complements of the possibly fractional ideals of the valuation ring and satisfying certain natural conditions if k is a henselian field of characteristic 0 or more generally an algebraically maximal kaplansky field we give an intrinsic construction of such a family which does not rely on a given truncation closed embedding we also show that towers of complements and truncation closed embeddings can be extended from an arbitrary field to at least one of its maximal immediate extensions

on the integrated density of states for crystals with randomly distributed impurities in the present paper we discuss spectral properties of a periodic schrodinger operator which is perturbed by randomly distributed impurities such operators occur as simple models for crystals or semi conductors with impurities while the spectrum itself is independent of the concentrationp of impurities for 0 p 1 we focus our attention on the limiting behavior of the integrated density of states p of the random schrodinger operator inside a spectral gap of the periodic operator asp 0 denoting byu0 the set of eigenvalues in the gap of the reference problem having precisely one impurity located at the origin say we show that the integrated density of states concentrates around the points ofu0 in the sense that p ue is of orderp for any fixed e neighborhoodue ofu0 while p k c p2 for any compact subsetk of the gap which does not intersectue

asymptotic behavior of the integrated density of states of acoustic operators with random long range perturbations in this paper we study the behavior of the integrated density of states of random acoustic operators of the form a 1 when is considered as an anderson type long range perturbations of some periodic function the behavior of the integrated density of states of a in the vicinity of the internal spectral edges is given

the large time asymptotics of some wiener integrals and the interband light absorption coefficient in the deep fluctuation spectrum in this paper we prove the existence of the interband light absorption coefficient and investigate its asymptotics for a number of models

periodic jacobi matrices on trees abstract we begin the systematic study of the spectral theory of periodic jacobi matrices on trees including a formal definition the most significant result that appears here for the first time is that these operators have no singular continuous spectrum we review important previous results of sunada and aomoto and present several illuminating examples we present many open problems and conjectures that we hope will stimulate further work

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

universality limits of a reproducing kernel for a half line schrodinger operator and clock behavior of eigenvalues we extend some recent results of lubinsky levin simon and totik from measures with compact support to spectral measures of schrodinger operators on the half line in particular we define a reproducing kernel sl for schrodinger operators and we use it to study the fine spacing of eigenvalues in a box of the half line schrodinger operator with perturbed periodic potential we show that if solutions u x are bounded in x by e epsilon x uniformly for near the spectrum in an average sense and the spectral measure is positive and absolutely continuous in a bounded interval i in the interior of the spectrum with xi 0 in i then uniformly in i frac s l xi 0 a l xi 0 b l s l xi 0 xi 0 rightarrow frac sin pi rho xi 0 a b pi rho xi 0 a b where d is the density of states we deduce that the eigenvalues near 0 in a large box of size l are spaced asymptotically as frac 1 l rho we adapt the methods used to show similar results for orthogonal polynomials

sum rules and spectral measures of schrodinger operators with l 2 potentials necessary and sufficient conditions are presented for a positive measure to be the spectral measure of a half line schrodinger operator with square integrable potential

hardy type derivations on fields of exponential logarithmic series abstract we consider the valued field k r of formal series with real coefficients and monomials in a totally ordered multiplicative group we investigate how to endow k with a logarithm l which satisfies some natural properties such as commuting with infinite products of monomials we studied derivations on k kuhlmann and matusinski in press km10 here we investigate compatibility conditions between the logarithm and the derivation i e when the logarithmic derivative is the derivative of the logarithm we analyze sufficient conditions on a given derivation to construct a compatible logarithm via integration of logarithmic derivatives in kuhlmann 2000 kuh00 the first author described the exponential closure k el of k l here we show how to extend such a log compatible derivation on k to k el

local fluctuation of the spectrum of a multidimensional anderson tight binding model we consider the anderson tight binding modelh v acting inl 2 z d and its restrictionh to finite hypercubes z d herev v x x z d is a random potential consisting of independent identically distributed random variables let e j j be the eigenvalues ofh and let j e e j e j 1 be its rescaled eigenvalues then assuming that the exponential decay of the fractional moment of the green function holds for complex energies neare and that the density of statesn e exists ate we shall prove that the random sequence j e j considered as a point process onr 1 converges weakly to the stationary poisson point process with intensity measuren e dx as gets large thus extending the result of molchanov proved for a one dimensional continuum random schrodinger operator on the other hand the exponential decay of the fractional moment of the green function was established recently by aizenman molchanov and graf as a technical lemma for proving anderson localization at large disorder or at extreme energy thus our result in this paper can be summarized as follows near the energye where anderson localization is expected there is no correlation between eigenvalues ofh if is large

the local structure of the spectrum of the one dimensional schrodinger operator let h v frac d text 2 dt text 2 q t omega be an one dimensional random schrodinger operator in 2 v v with the classical boundary conditions the random potentialq t has a formq t f x t wherex t is a brownian motion on the compact riemannian manifoldk andf k r 1 is a smooth morse function mathop min limits k f 0 let n v delta sum limits ei v in delta 1 where 0 e i v are the eigenvalues ofh v the main result theorem 1 of this paper is the following ifv e 0 0 k z anda 0 a is a fixed constant then p left n v left e 0 frac a 2v e 0 frac a 2v right k right xrightarrow v to infty e an e 0 an e 0 k k wheren e 0 is a limit state density ofh v v this theorem mean that there is no repulsion between energy levels of the operatorh v v the second result theorem 2 describes the phenomen of the repulsion of the corresponding wave functions

a criterion for completeness of casimir operators we give a criterion for a set of casimir operators of any semisimple lie algebra to be an algebraically independent generating set of the algebra of the casimir operators the criterion is applied to supply complete sets for the casimir operators of al bl cl and dl with the aid of a method to construct some casimir operators we then furnish complete sets for the casimir operators of the exceptional lie algebras using the criterion

large deviations and lifshitz singularity of the integrated density of states of random hamiltonians we consider the integrated density of states ids of random hamiltonian h v v being a random field on d which satisfies a mixing condition we prove that the probability of large fluctuations of the finite volume ids 1 h d around the thermodynamic limit is bounded from above by exp k k 0 in this case can be recovered from a variational principle furthermore we show the existence of a lifshitztype of singularity of as 0 in the case where v is non negative more precisely we prove the following bound exp k d 2 as 0 k 0 this last result is then discussed in some examples

special energies and special frequencies special frequencies have been asserted to be zeros of the density of frequencies corresponding to a random chain of coupled oscillators our investigation includes both this model and the random one dimensional schrodinger operator describing an alloy or its discrete analogue using the phase method we exactly determine a bilateral lifsic asymptotic of the integrated density of statesk e at special energiese s which is not only of the classical type exp c e e s 1 2 but also exp c e e s is a typical behaviour in addition other asymptotics occur e g e e s c which show thatk e need not bek

franco american meeting on the mathematics of random and almost periodic potentials we present the abstracts of papers given at a conference held at the ecole polytechnique in june 1983 on the mathematics of random and almost periodic potentials

linear wegner estimate for alloy type schrodinger operators on metric graphs we study spectra of alloy type random schrodinger operators on metric graphs for finite edge subsets we prove a wegner estimate which is linear in the volume i e the total length of the edges and the length of the energy interval the single site potential needs to have fixed sign the metric graph does not need to have a periodic structure a further result is the existence of the integrated density of states for ergodic random hamiltonians on metric graphs with a z structure for certain models the two above results together imply the lipschitz continuity of the integrated density of states

eigenvalue branches of the schrodinger operatorh λw in a gap of σ h the authors study the eigenvalue branches of the schrodinger operatorh w in a gap of h in particular they consider questions of asymptotic distribution of eigenvalues and bounds on the number of branches they also address the completeness problem

localization of the continuous anderson hamiltonian in 1 d we study the bottom of the spectrum of the anderson hamiltonian mathcal h l partial x 2 xi on 0 l driven by a white noise xi and endowed with either dirichlet or neumann boundary conditions we show that as l rightarrow infty the point process of the appropriately shifted and rescaled eigenvalues converges to a poisson point process on mathbf r with intensity e x dx and that the appropriately rescaled eigenfunctions converge to dirac masses located at independent and uniformly distributed points furthermore we show that the shape of each eigenfunction recentered around its maximum and properly rescaled is given by the inverse of a hyperbolic cosine we also show that the eigenfunctions decay exponentially from their localization centers at an explicit rate and we obtain very precise information on the zeros and local maxima of these eigenfunctions finally we show that the eigenvalues eigenfunctions in the dirichlet and neumann cases are very close to each other and converge to the same limits

fine structure of the zeros of orthogonal polynomials iv a priori bounds and clock behavior we prove locally uniform spacing for the zeros of orthogonal polynomials on the real line under weak conditions jacobi parameters approach the free ones and are of bounded variation we prove that for ergodic discrete schrodinger operators poisson behavior implies a positive lyapunov exponent both results depend on a priori bounds on eigenvalue spacings for which we provide several proofs

on the largest singular values of random matrices with independent cauchy entries we apply the method of determinants to study the distribution of the largest singular values of large m n real rectangular random matrices with independent cauchy entries we show that for a special one parametric class of statistics the properties of the largest singular values rescaled by a factor 1 m2n2 agree in the limit with the statistical properties of the poisson random point process with the intensity 1 x 3 2 and therefore are different from the tracy widom law among other corollaries of our method we show an interesting connection between the mathematical expectations of the determinants of the complex rectangular m n standard wishart ensemble and the real rectangular 2m 2n standard wishart ensemble

uniform existence of the integrated density of states for random schrodinger operators on metric graphs over zd abstract we consider ergodic random schrodinger operators on the metric graph z d with random potentials and random boundary conditions taking values in a finite set we show that normalized finite volume eigenvalue counting functions converge to a limit uniformly in the energy variable this limit the integrated density of states can be expressed by a closed shubin pastur type trace formula it supports the spectrum and its points of discontinuity are characterized by existence of compactly supported eigenfunctions among other examples we discuss random magnetic fields and percolation models

concentration of eigenvalues for skew shift schrodinger operators i analyze the microscopic behavior of the eigenvalues of skew shift schrodinger operators and show that their statistics must resemble the one of the anderson model rather than the one of quasi periodic schrodinger operators

on the joint distribution of energy levels of random schrodinger operators we consider operators with random potentials on graphs such as the lattice version of the random schrodinger operator the main result is a general bound on the probabilities of simultaneous occurrence of eigenvalues in specified distinct intervals with the corresponding eigenfunctions being separately localized within prescribed regions the bound generalizes the wegner estimate on the density of states the analysis proceeds through a new multi parameter spectral averaging principle

brownian survival and lifshitz tail in perturbed lattice disorder abstract we consider the annealed asymptotics for the survival probability of brownian motion among randomly distributed traps the configuration of the traps is given by independent displacements of the lattice points we determine the long time asymptotics of the logarithm of the survival probability up to a multiplicative constant as applications we show the lifshitz tail effect of the density of states of the associated random schrodinger operator and derive a quantitative estimate for the strength of intermittency in the parabolic anderson problem

the repulsion between localization centers in the anderson model in this note we show that a simple combination of deep results in the theory of random schrodinger operators yields a quantitative estimate of the fact that the localization centers become far apart as corresponding energies are close together

a complete basis set of functionally independent invariants of continuous lie groups using the methods of differential equations the author has obtained the complete set of functionally independent bases of invariants for some continuous lie groups of mathematical physics i the conformal group ii the semi direct product of the weyl group and the one parameter group of psi translation iii the direct product of the poincare group and the one parameter group of psi scaling iv the weyl group v the 14 parameter maximal symmetry group of the schrodinger equation for the coulombic system of atoms and molecules and vi su 3 of elementary particle physics the first four groups are the only groups which together with the poincare group are the maximal symmetry groups of all forms of the non linear klein gordon equation

a diamagnetic inequality for semigroup differences the diamagnetic inequality for the magnetic schrodinger semigroup is extended to the difference of the semigroups of magnetic schrodinger operators with neumann and dirichlet boundary conditions on arbitrary open domains and rather general magnetic vector potentials a and potentials v in particular this bound renders moot all the technical issues in the recent proofs of the independence of the boundary conditions for the integrated density of states for magnetic schrodinger operators independence of the boundary conditions for the free case that is for vanishing potentials and vector potentials immediately implies independence of the boundary conditions of the integrated density of states for a large class of magnetic schrodinger operators

bounds on the density of states for schrodinger operators we establish bounds on the density of states measure for schrodinger operators these are deterministic results that do not require the existence of the density of states measure or equivalently of the integrated density of states the results are stated in terms of a density of states outer measure that always exists and provides an upper bound for the density of states measure when it exists we prove log holder continuity for this density of states outer measure in one two and three dimensions for schrodinger operators and in any dimension for discrete schrodinger operators

eigenvalue statistics for lattice hamiltonian with off diagonal disorder this short note deals with a certain kind of lattice hamiltonian with off diagonal disorder based on the exponential decay of the fractional moment of the green function we are able to prove that the properly rescaled eigenvalues of the random hamiltonian are distributed as a poisson point process with intensity measure given by the density of states one of the key step in this proof is the minami type estimate as a crucial ingredient we also use the minami type estimate to study some important properties of the random hamiltonian such as multiplicity of the eigenvalues and quantitative estimate of the localization centers

on the basic states of one dimensional disordered structures the purpose of this paper is to study a limit probability distribution of the set of the first eigenvalues 1 2 with a fixed and of the boundary problem on the interval 0 wherea t q t are the random stationary processes particularly the question of the repulsion between the first eigenvalues small energetic levels is studied it has been proved that in the divergent case q t 0 a t 0 levels repulsion exists as for the potential case a t 1 q t 0 there is not any repulsion at all this is one of the main differences between these two cases

on the lie symmetries of the classical kepler problem the standard version of noether s theorem when applied to the classical kepler problem leads to the constants of energy and angular momentum but does not give the hidden symmetry known as the runge lenz vector lie s theory of differential equations is used to obtain all three constants of motion the transformations of solutions under the point transformations to which these constants correspond are studied the results are generalised to n dimensions

tracy widom at high temperature we investigate the marginal distribution of the bottom eigenvalues of the stochastic airy operator when the inverse temperature beta tends to 0 we prove that the minimal eigenvalue whose fluctuations are governed by the tracy widom beta law converges weakly when properly centered and scaled to the gumbel distribution more generally we obtain the convergence in law of the marginal distribution of any eigenvalue with given index k those convergences are obtained after a careful analysis of the explosion times process of the riccati diffusion associated to the stochastic airy operator we show that the empirical measure of the explosion times converges weakly to a poisson point process using estimates proved in dumaz and virag ann inst h poincare probab statist 49 4 915 933 2013 we further compute the empirical eigenvalue density of the stochastic airy ensemble on the macroscopic scale when beta rightarrow 0 as an application we investigate the maximal eigenvalues statistics of beta n ensembles when the repulsion parameter beta n rightarrow 0 when n rightarrow infty we study the double scaling limit n rightarrow infty beta n rightarrow 0 and argue with heuristic and numerical arguments that the statistics of the marginal distributions can be deduced following the ideas of edelman and sutton j stat phys 127 6 1121 1165 2007 and ramirez et al j am math soc 24 919 944 2011 from our later study of the stochastic airy operator

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

the effect of boundaries on the spectrum of a one dimensional random mass dirac hamiltonian the average density of states dos of the one dimensional dirac hamiltonian with a random mass on a finite interval 0 l is derived our method relies on the eigenvalue distributions extreme value statistics problem which are obtained explicitly the well known dyson singularity is recovered above the crossover energy below c we find a log normal suppression of the average dos

random hamiltonians ergodic in all but one direction letv 1 andv 2 be two ergodic random potentials on d we consider the schrodinger operatorh h0 v withh0 and forx x1 x2 v omega x left begin array 20 c v omega 1 x if x 1 over smash scriptstyle vphantom x 0 end array right we prove certain ergodic properties of the spectrum for this model and express the integrated density of states in terms of the density of states of the stationary potentialsv 1 andv 2 finally we prove the existence of the density of surface states forh

the central limit theorem for the spectrum of the random one dimensional schrodinger operator let hl d2 dt2 q t be an one dimensional random schrodinger operator in 2 l l with the classical boundary conditions the random potential q t has a form q t f xt where xt is a brownian motion on the euclidean v dimensional torus f sv r1 is a smooth function with the nondegenerated critical points minsv f 0 let n l lambda sum lambda i l leqslant lambda 1 lambda i l omega are the eigenvalues of hl be a spectral distribution function in the volume l l and n liml 1 2l nl be a corresponding limit distribution function

the density of states in the anderson model at weak disorder a renormalization group analysis of the hierarchical model we study the density of states in a hierarchical approximation of the anderson tight binding model at weak disorder using a renormalization group approach since the laplacian term in our model is hierarchical the renormalization group transformations act essentially on the local potential distribution and the energy technically we use the supersymmetric replica trick and study the averaged green s function starting with a gaussian distribution with small variance we find that the density of states is analytic as soon as the variance of the potential is turned on except possibly near the band edge where we can show this only for 2 which corresponds tod 4 moreover it is perturbatively close to the free one except near the eigenvalues of the hierarchical laplacian where it is given up to perturbative corrections by the rescaled potential distribution

on the density of states of schrodinger operators with a random potential using very recent results on ergodic theorems for superadditive processes on rd the authors prove the existence of the density of states for a wide class of random schrodinger operators in particular new non asymptotic estimates on the density of states are obtained and examples are discussed

statistical properties of spectral fluctuations for a quantum system with infinitely many components extending the idea formulated in makino phys rev e 67 066205 2003 that is based on the berry robnik approach m v berry and m robnik j phys a 17 2413 1984 we investigate the statistical properties of a two point spectral correlation for a classically integrable quantum system the eigenenergy sequence of this system is regarded as a superposition of infinitely many independent components in the semiclassical limit we derive the level number variance lnv in the limit of infinitely many components and discuss its deviations from poisson statistics the slope of the limiting lnv is found to be larger than that of poisson statistics when the individual components have a certain accumulation this property agrees with the result from the semiclassical periodic orbit theory that is applied to a system with degenerate torus actions d biswas m azam and s v lawande phys rev a 43 5694 1991

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

lifschitz tail and wiener sausage ii the present paper is a continuation of 23 we present here two examples of situations where the method developed in 23 applies the first example is that of a periodic diffusion z in r n 1 with generator l isia dj where a is a c2 z periodic elliptic matrix we consider the wiener sausage w u0 c u i z c where c is a nonpolar compact subset of r for instance a closed ball or disc in r if a denotes the constant matrix of homogenized coefficients of the periodic diffusion see 1 i we show that

correlation estimates in the anderson model we give a new proof of correlation estimates for arbitrary moments of the resolvent of random schrodinger operators on the lattice that generalizes and extends the correlation estimate of minami for the second moment we apply this moment bound to obtain a new n level wegner type estimate that measures eigenvalue correlations through an upper bound on the probability that a local hamiltonian has at least n eigenvalues in a given energy interval another consequence of the correlation estimates is that the results on the poisson statistics of energy level spacing and the simplicity of the eigenvalues in the strong localization regime hold for a wide class of translation invariant selfadjoint lattice operators with decaying off diagonal terms and random potentials

the symmetry and some exact solutions of the nonlinear many dimensional liouville d alembert and eikonal equations multiparametrical exact solutions of the many dimensional nonlinear d alembert liouville sine gordon and eikonal equations are obtained the maximally extensive local invariance groups of the equations are determined and invariants of the extended poincare group are found

lifschitz tails for the anderson model we present in an expository way an elementary rigorous proof patterned after an argument of kirsch martinelli that the anderson model has lifschitz tails in very great generality

breaking supersymmetry in a one dimensional random hamiltonian the one dimensional supersymmetric random hamiltonian where x is a gaussian white noise of zero mean and variance g presents particular spectral and localization properties at low energy a dyson singularity in the integrated density of states idos n e 1 ln2e and a delocalization transition related to the behavior of the lyapunov exponent inverse localization length vanishing like e 1 ln e as e 0 we study how this picture is affected by breaking supersymmetry with a scalar random potential h hsusy v x where v x is a gaussian white noise of variance in the limit g3 a fraction of states n 0 g ln2 g3 migrate to the negative spectrum and the lyapunov exponent reaches a finite value 0 g ln g3 at e 0 the exponential lifshits tail of the idos for e is studied in detail and is shown to involve a competition between the two noises x and v x whatever the larger is this analysis relies on analytic results for n e and e obtained by two different methods a stochastic method and the replica method the problem of extreme value statistics of eigenvalues is also considered distribution of the nth excited state energy the results are analyzed in the context of classical diffusion in a random force field in the presence of random annihilation creation local rates

eigenvalue statistics for schrodinger operators with random point interactions on rd d 1 2 3 we prove that the local eigenvalue statistics at energy e in the localization regime for schrodinger operators with random point interactions on rd for d 1 2 3 is a poisson point process with the intensity measure given by the density of states at e times the lebesgue measure this is one of the first examples of poisson eigenvalue statistics for the localization regime of multi dimensional random schrodinger operators in the continuum the special structure of resolvent of schrodinger operators with point interactions facilitates the proof of the minami estimate for these models

localization and spectral statistics in a banded random matrix ensemble we investigate the localization properties of the eigenvectors of a banded random matrix ensemble in which the diagonal matrix elements increase along the diagonal we relate the results to a transition in the spectral statistics which is observed as a parameter is vaned and discuss the relevance of this model to the quantum mechanics of chaotic hamiltonian systems

global bounds for the lyapunov exponent and the integrated density of states of random schrodinger operators in one dimension in this paper we prove an upper bound for the lyapunov exponent e and a two sided bound for the integrated density of states n e at an arbitrary energy e 0 of random schrodinger operators in one dimension these schrodinger operators are given by potentials of identical shape centred at every lattice site but with non overlapping supports and with randomly varying coupling constants both types of bounds only involve scattering data for the single site potential they show in particular that both e and n e e 1 2 decay at infinity at least like 1 e 1 2 as an example we consider the random kronig penney model

individual energy level distributions for one dimensional diagonal and off diagonal disorder we study the distribution of the n th energy level for two different one dimensional random potentials this distribution is shown to be related to the distribution of the distance between two consecutive nodes of the wave function we first consider the case of a white noise potential and study the distributions of energy level both in the positive and the negative part of the spectrum it is demonstrated that in the limit of a large system l the distribution of the n th energy level is given by a scaling law which is shown to be related to the extreme value statistics of a set of independent variables in the second part we consider the case of a supersymmetric random hamiltonian potential v x x 2 x we study first the case of x being a white noise with zero mean it is in particular shown that the ground state energy which behaves on average like exp l 1 3 in agreement with previous work is not a self averaging quantity in the limit l as is seen in the case of diagonal disorder then we consider the case when x has a non zero mean value

smoothness of the density of states in the anderson model at high disorder we prove smoothness of the density of states in the anderson model at high disorder for a class of potential distributions that include the uniform distribution