wegner estimate for gaussian random magnetic fields for the schrodinger operator on l2 r2 with the magnetic field which is a sample path of a stationary gaussian random field a wegner type estimate applicable for the proof of the anderson localization is proven by referring a recent method by erd s and hasler and the theory of the malliavin calculus

induced i l fuzzy topological spaces abstract in this paper we introduce the notion of induced i l fuzzy topological spaces by using the i l valued lower semicontinuous mappings and investigate some of their basic properties in particular we give a necessary and sufficient condition for an i l fuzzy topological space to be induced and prove that this kind of induced space preserves the cartesian product and the n compactness as well

low temperature behavior of the multicomponent widom rowlison model on finite square lattices we consider the multicomponent widom rowlison with metropolis dynamics which describes the evolution of a particle system where m different types of particles interact subject to certain hard core constraints focusing on the scenario where the spatial structure is modeled by finite square lattices we study the asymptotic behavior of this interacting particle system in the low temperature regime analyzing the tunneling times between its m maximum occupancy configurations and the mixing time of the corresponding markov chain in particular we develop a novel combinatorial method that exploiting geometrical properties of the widom rowlinson configurations on finite square lattices leads to the identification of the timescale at which transitions between maximum occupancy configurations occur and shows how this depends on the chosen boundary conditions and the square lattice dimensions

lifshitz tail for schrodinger operator with random magnetic field we study the behavior of the density states at the lower edge of the spectrum for schrodinger operators with random magnetic fields we use a new estimate on magnetic schrodinger operators which is similar to the avron herbst simon estimate but the bound is always nonnegative

on the lipschitz continuity of the integrated density of states for sign indefinite potentials the present paper is devoted to the study of spectral properties of random schrodinger operators using a finite section method for toeplitz matrices we prove a wegner estimate for some alloy type models where the single site potential is allowed to change sign the results apply to the corresponding discrete model too in certain disorder regimes we are able to prove the lipschitz continuity of the integrated density of states and or localization near spectral edges

a remark on the dirichlet neumann decoupling and the integrated density of states abstract we prove an estimate on the difference of the number of eigenvalues for schrodinger operators with dirichlet and neumann boundary conditions in large boxes in the proof we use krein s theory of the spectral shift function as an application we show that the integrated density of states is independent of the choice of boundary conditions

the uniqueness of the integrated density of states for the schrodinger operators with magnetic fields the integrated density of states ids for the schrodinger operators is defined in two ways by using the counting function of eigenvalues of the operator restricted to bounded regions with appropriate boundary conditions or by using the spectral projection of the whole space operator a sufficient condition for the coincidence of the two definitions above is given moreover a sufficient condition for the coincidence of the ids for the dirichlet boundary conditions and the ids for the neumann boundary conditions is given the proof is based only on the fundamental items in functional analysis such as the min max principle etc

induced i l fuzzy topological vector spaces abstract in this paper we introduce and study the induced i l fuzzy topological vector spaces the following main result is proved if l x is an l fuzzy topological vector space then i l x i l is an i l fuzzy topological vector space where i l is the fuzzy unit interval and i l is the induced i l fuzzy topology of the l fuzzy topology furthermore we also prove that the induced i l fuzzy topological vector spaces i l x i l preserve the product and quotient space as well

non monotonic random schrodinger operators the anderson model abstract we prove exponential localization at all energies for one dimensional continuous anderson type models with single site potentials of changing sign a periodic background potential is allowed the main problem arises from non monotonicity i e the operator does not depend monotonically in the form sense on the random parameters we show that the method of two parameter spectral averaging recently devised by buschmann and stolz to prove localization for poisson and random displacement models can be modified to work for the type of anderson model considered here

phase transitions in the multicomponent widom rowlinson model and in hard cubes on the bcc lattice we use monte carlo techniques and analytical methods to study the phase diagram of the m component widom rowlinson model on the bcc lattice there are m species all with same fugacity z and a nearest neighbor hard core exclusion between unlike particles simulations show that for m 3 there is a crystal phase for z lying between zc m and zd m while for z zd m there are m demixed phases each consisting mostly of one species for m 2 there is a direct second order transition from the gas phase to the demixed phase while for m 3 the transition at zd m appears to be first order putting it in the potts model universality class for m large pirogov sinai theory gives zd m m 2 2 3m2 in the crystal phase the particles preferentially occupy one of the sublattices independent of species i e spatial symmetry but not particle symmetry is broken for m this transition approaches that of the one component hard cube gas with fugacity y zm we find by direct simulations of such a system a transition at yc 0 71 which is consistent with the simulation zc m for large m this transition appears to be always of the ising type

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

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

generalization of kolmogoroff s theorem to l topological vector spaces in this paper the concept of l fuzzy normed linear spaces is introduced and some of its properties are studied then we give a necessary and sufficient condition for l topological vector spaces to be l fuzzy normable this is a generalization of kolmogoroff s theorem in the classical topological vector spaces to l topological vector spaces

low lying eigenvalues of randomly curved quantum waveguides abstract we consider the negative dirichlet laplacian on an infinite waveguide embedded in r 2 and finite segments thereof the waveguide is a perturbation of a periodic strip in terms of a sequence of independent identically distributed random variables which influence the curvature we derive explicit lower bounds on the first eigenvalue of finite segments of the randomly curved waveguide in the small coupling i e weak disorder regime this allows us to estimate the probability of low lying eigenvalues a tool which is relevant in the context of anderson localization for random schrodinger operators

local convexity and local boundedness of induced i l topological vector spaces in this paper we prove that an l topological vector space is locally convex locally bounded iff its induced i l topological vector space is locally convex locally bounded where i l is the l fuzzy unit interval also we give a necessary and sufficient condition for the induced i l topological vector space to be i l fuzzy normable

asymptotic ergodicity of the eigenvalues of random operators in the localized phase we prove that for a general class of random operators the family of the unfolded eigenvalues in the localization region is asymptotically ergodic in the sense of minami spectra of random operators and related topics 2011 minami conjectured this to be the case for discrete anderson model in the localized regime we also provide a local analogue of this result from the asymptotics ergodicity one can recover the statistics of the level spacings as well as a number of other spectral statistics our proofs rely on the analysis developed in germinet and klopp spectral statistics for random schrodinger operators in the localized regime 2010

separation properties of induced i l topological spaces the concept of induced i l topological spaces has been introduced by kubiak ph d thesis uam poznan 1985 and independently by wang kexue tongbao 34 5 1989 333 in this paper the separation properties in the sense of hutton reilly of induced i l topological spaces are investigated the main result of the paper is a characterization of l topological spaces by means of the appropriate hutton reilly separation properties of its induced i l topological space

wegner estimate and anderson localization for random magnetic fields we consider a two dimensional magnetic schrodinger operator with a spatially stationary random magnetic field we assume that the magnetic field has a positive lower bound and that it has fourier modes on arbitrarily short scales we prove the wegner estimate at arbitrary energy i e we show that the averaged density of states is finite throughout the whole spectrum we also prove anderson localization at the bottom of the spectrum

nonmonotonic behavior in hard core and widom rowlinson models we give two examples of nonmonotonic behavior in symmetric systems exhibiting more than one critical point at which spontaneous symmetry breaking appears or disappears the two systems are the hard core model and the widom rowlinson model and both examples take place on a variation of the cayley tree bethe lattice devised by schonmann and tanaka we obtain similar though less constructive examples of non monotonicity via certain local modifications of any graph e g the square lattice which is known to have a critical point for either model en route we prove that the widom rowlinson model does behave monotonically on the ordinary cayley tree and we obtain an explicit description of its critical behavior there we conclude with some results about monotonicity of the phase transition phenomenon relative to graph structure

an alternate version of pirogov sinai theory a new approach to the pirogov sinai theory of phase transitions is developed not employing the contour models with a parameter the completeness of the phase diagram is proven

ordering and demixing transitions in multicomponent widom rowlinson models we use monte carlo techniques and analytical methods to study the phase diagram of multicomponent widom rowlinson models on a square lattice there are m species all with the same fugacity z and a nearest neighbor hard core exclusion between unlike particles simulations show that for m between two and six there is a direct transition from the gas phase at z mathit z mathit d m to a demixed phase consisting mostly of one species at zg mathit z mathit d m while for m ensuremath ge 7 there is an intermediate crystal phase for z lying between mathit z mathit c m and mathit z mathit d m in this phase which is driven by entropy particles independent of species preferentially occupy one of the sublattices i e spatial symmetry but not particle symmetry is broken the transition at mathit z mathit d m appears to be of the first order for m ensuremath ge 5 putting it in the potts model universality class for large m the transition between the crystalline and demixed phases at mathit z mathit d m can be proven to be of the first order with mathit z mathit d m ensuremath sim m 2 1 m while mathit z mathit c m is argued to behave as ensuremath lambda mathrm cr m with ensuremath lambda mathrm cr the critical value of the fugacity at which the one component hard core square lattice gas has a transition and to be always of the ising type explicit calculations for the bethe lattice with the coordination number q 4 give results similar to those for the square lattice except that the transition at mathit z mathit d m becomes first order at mg2 this happens for all q consistent with the model being in the potts universality class c 1995 the american physical society

optimal volume wegner estimate for random magnetic laplacians on z2 we consider a two dimensional magnetic schrodinger operator on a square lattice with a spatially stationary random magnetic field we prove a wegner estimate with optimal volume dependence the wegner estimate holds around the spectral edges and it implies holder continuity of the integrated density of states in this region the proof is based on the wegner estimate obtained in erd s and hasler wegner estimate for random magnetic laplacians on z2 ann henri poincare 12 1719 1731 2012 10 1007 s00023 012 0177 9

localization of classical waves ii electromagnetic waves we consider electromagnetic waves in a medium described by a position dependent dielectric constant we assume that is a random perturbation of a periodic function and that the periodic maxwell operator has a gap in the spectrum where we prove the existence of localized waves i e finite energy solutions of maxwell s equations with the property that almost all of the wave s energy remains in a fixed bounded region of space at all times localization of electromagnetic waves is a consequence of anderson localization for the self adjoint operators we prove that in the random medium described by the random operator exhibits anderson localization inside the gap in the spectrum of this is shown even in situations when the gap is totally filled by the spectrum of the random operator we can prescribe random environments that ensure localization in almost the whole gap

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

an integral equation theory for the widom rowlinson mixture the widom rowlinson mixture is a two component fluid in which like species do not interact and unlike species interact via a hard core repulsion as the density is increased this fluid phase separates standard integral equation approaches such as the percus yevick or hypernetted chain or thermodynamically self consistent hybirds of these two make very inaccurate predictions for the location of this critical point in the three dimensional model in this article we suggest a family of new approximations for this model that rely on incorporating terms in the density expansion of the direct correlation function into the closure approximation we show that the simplest of these closures is significantly more accurate than previous theories for the structure and thermodynamics of the fluid

the lp theory of the spectral shift function the wegner estimate and the integrated density of states for some random operators we develop the l p theory of the spectral shift function for p 1 applicable to pairs of self adjoint operators whose difference is in the trace ideal p for 0 p 1 this result is a key ingredient of a new short proof of the wegner estimate applicable to a wide variety of additive and multiplicative random perturbations of deterministic background operators the proof yields the correct volume dependence of the upper bound this implies the local holder continuity of the integrated density of states at energies in the unperturbed spectral gap under an additional condition of the single site potential local holder continuity is proved at all energies this new wegner estimate together with other standard results establishes exponential localization for a new family of models for additive and multiplicative perturbations

on i l topological vector spaces generated by a co tower of l topological vector spaces in this paper a characterization for an i l topological space to be generated by a given co tower of l topological spaces is obtained moreover the relationship between some properties of an i l topological vector space generated by a co tower of l topological vector spaces and the corresponding properties of the given co tower of l topological vector spaces is investigated our results show that if an i l topological vector space generated by a co tower of l topological vector spaces has some properties such as local convexity and local boundedness then all l topological vector spaces in the co tower also have the same properties but the converse is incorrect even in the case of i topological vector space generated by a co tower of classical topological vector spaces finally we supply a necessary and sufficient condition for an i l topological vector space generated by a co tower of l topological vector spaces with some properties such as local convexity and local boundedness to have such properties too

anderson localization for random schrodinger operators with long range interactions we prove pure point spectrum with exponentially decaying eigenfunctions at all band edges for schrodinger operators with a periodic potential plus a random potential of the form v w x q i w f x i where f decays at infinity like x m for m 4d resp m 3d depending on the regularity of f the random variables q i are supposed to be independent and identically distributed we assume that their distribution has a bounded density of compact support

localization in general one dimensional random systems we discuss two ways of extending the recent ideas of localization from discrete schrodinger operators jacobi matrices to the continuum case one case allows us to prove localization in the goldshade molchanov pastur model for a larger class of functions than previously the other method studies the model v wherev is a random constant in each hyper cube we extend wegner s result on the lipschitz nature of the ids to this model

l fuzzy normal spaces and tietze extension theorem abstract in this paper we prove a characterization theorem for normal l fuzzy topological spaces l is an infinitely distributive lattice with an order reversing involution in the particular case l 0 1 this theorem reduces to a known result of kat tov fund math 38 1951 85 91 and tong duke math j 19 1952 289 292 as an important application we obtain a fuzzy version of tietze extension theorem this yields an affirmative answer to a question raised in a series of papers by rodabaugh fuzzy sets and systems 11 1983 163 183

higher separation axioms in l topologically generated i l topological spaces the main result of this paper is a theorem about inserting a pair of semicontinuous l real valued functions which extends the insertion theorem of kubiak comment math univ carolinae 34 1993 357 362 from l 0 1 to an arbitrary meet continuous lattice l endowed with an order reversing involution with this result it is shown that the normality type separation axioms in top l are preserved by the functor which takes an l topological space x to the i l topological space l x obtained by providing the set x with the i l topology consisting of all lower semicontinuous functions from x to i l the same is proved for the case of the regularity axiom

i l topological vector spaces and its level topological spaces in this paper the relationship between i l topological vector spaces and its level topological spaces is discussed the main result of this paper is to give a method which generates an i l topological vector space by a co tower of vector l topologies this result is a generalization of previous results published in this journal by the same authors

the integrated density of states for some random operators with nonsign definite potentials we study the integrated density of states of random anderson type additive and multiplicative perturbations of deterministic background operators for which the single site potential does not have a fixed sign our main result states that under a suitable assumption on the regularity of the random variables the integrated density of states of such random operators is locally holder continuous at energies below the bottom of the essential spectrum of the background operator for any nonzero disorder and at energies in the unperturbed spectral gaps provided the randomness is sufficiently small the result is based on a proof of a wegner estimate with the correct volume dependence the proof relies upon the lp theory of the spectral shift function for p 1 comm math phys 218 2001 113 130 and the vector field methods of klopp comm math phys 167 1995 553 569 we discuss the application of this result to schrodinger operators with random magnetic fields and to band edge localization

localization of classical waves i acoustic waves we consider classical acoustic waves in a medium described by a position dependent mass density x we assume that x is a reandom perturbation of a periodic function 0 x and that the periodic acoustic operator a 0 nabla cdot tfrac 1 varrho 0 x nabla has a gap in the spectrum we prove the existence of localized waves i e finite energy solutions of the acoustic equations with the property that almost all of the wave s energy remains in a fixed bounded region of space at all times with probability one localization of acoustic waves is a consequence of anderson localization for the self adjoint operators a nabla cdot tfrac 1 varrho 0 x nabla onl2 d we prove that in the random medium described by x the random operatora exhibits anderson localization inside the gap in the spectrum ofa0 this is shown even in situations when the gap is totally filled by the spectrum of the random opertor we can prescribe random environments that ensure localization in almost the whole gap

localization for some continuous random hamiltonians in d dimensions abstract we prove the existence with probability one of an interval of pure point spectrum for some families of continuous random schrodinger operators in d dimensions for anderson like models with positive short range single site potentials we also prove that the corresponding eigenfunctions decay exponentially and that the integrated density of states is lipschitz continuous for the other families of random potentials that we study we show that the corresponding eigenfunctions decay faster than an inverse power of x which depends upon the decay rate of the single site potential to obtain these results we develop an extension of the classical aronszajn donoghue theory for a class of relatively compact perturbations and a spectral averaging method which extends kotani s trick to these more general families of operators

localization for some continuous random schrodinger operators we study the spectrum of random schrodinger operators acting onl2 r d of the following type h delta w sum x in mathbb z d t x v x the t x x in mathbb z d are i i d random variables under weak assumptions onv we prove exponential localization forh at the lower edge of its spectrum in order to do this we give a new proof of the wegner estimate that works without sign assumptions onv

locally bounded l topological vector spaces the purpose of this paper is to introduce the concept of locally bounded l topological vector spaces and prove that each locally bounded l vector topology on x can be determined by an l fuzzy quasi norm

inverse tunnelingestimates and applications to the study of spectral statistics ofrandom operators on the real line we present a proof of a minami type estimate for one dimensional random schrodinger operators valid at all energies in the localization regime provided a wegner estimate is known to hold the minami type estimate is then applied to two models to obtain results on their spectral statistics the heuristics underlying our proof of minami type estimates is that close by eigenvalues of a one dimensional schrodinger operator correspond either to eigenfunctions that live far away from each other in space or they come from some tunneling phenomena in the second case one can undo the tunneling and thus construct quasi modes that live far away from each other in space resume nous demontrons une inegalite de type minami pour des operateurs de schrodinger aleatoires uni dimensionnel dans toute la region localisee si une estimee de wegner est connue cette estimee de type de minami est alors appliquee pour obtenir les statistiques spectrales pour deux modeles l heuristique qui guide ce travail est que des valeurs propres proches pour un operateur de schrodinger sur un intervalle sont soient localisees loin l une de l autre soit sont la consequence d un phenomene d effet tunnel dans le second cas on peut en defaisant cet effet tunnel construire des quasi modes qui sont localises loin l un de l autre 0 introduction consider the following two random operators on the real line the anderson model 0 1 h d2 dx2 w

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

bounds on the spectral shift function and the density of states we study spectra of schrodinger operators on d first we consider a pair of operators which differ by a compactly supported potential as well as the corresponding semigroups we prove almost exponential decay of the singular values n of the difference of the semigroups as n and deduce bounds on the spectral shift function of the pair of operators