integrable forms of the one dimensional flow equation for unsaturated heterogeneous porous media the equation for the horizontal transport of a liquid in an unsaturated scale heterogeneous porous medium is t x x c x x e x x c e d a systematic search for lie backlund symmetries leads to the requirement that c a b 2 as in the homogeneous 1 case more generally e may be 1 mx 1 3 2 c or exp mx 3c 2 in these cases the transport equation may be linearized and solved exactly examples of more complicated heterogeneous extensions are presented for the integrable nonlinear diffusion equations and for burgers equation

a short proof of zheludev s theorem we give a short proof of zheludev s theorem that states the existence of precisely one eigenvalue in sufficiently distant spectral gaps of a hill operator subject to certain short range perturbations as a by product we simultaneously recover rofe beketov s result about the finiteness of the number of eigenvalues in essential spectral gaps of the perturbed hill operator our methods are operator theoretic in nature and extend to other one dimensional systems such as perturbed periodic dirac operators and weakly perturbed second order finite difference operators we employ the trick of using a selfadjoint birman schwinger operator even in cases where the perturbation changes sign a method that has already been successfully applied in different contexts and appears to have further potential in the study of point spectra in essential spectral gaps

numerical computation of eigenvalues in spectral gaps of sturm liouville operators we consider two different approaches for the numerical calculation of eigenvalues of a singular sturm liouville problem y q x email protected email protected r where the potential q is a decaying l 1 perturbation of a periodic function and the essential spectrum consequently has a band gap structure both the approaches which we propose are spectrally exact they are capable of generating approximations to eigenvalues in any gap of the essential spectrum and do not generate any spurious eigenvalues we also prove theorem 2 4 that even the most careless of regularizations of the problem can generate at most one spurious eigenvalue in each spectral gap a result which does not seem to have been known hitherto

analytical and numerical solution of a generalized stefan problem exhibiting two moving boundaries with application to ocean delta formation abstract a model associated with the formation of sedimentary ocean deltas is presented this model is a generalized one dimensional stefan problem bounded by two moving boundaries the shoreline and the alluvial bedrock transition the sediment transport is a non linear diffusive process the diffusivity modeled as a power law of the fluvial slope dimensional analysis shows that the first order behavior of the moving boundaries is determined by the dimensionless parameter 0 r a b 1 the ratio of the fluvial slope to bedrock slope at the alluvial bedrock transition a similarity form of the governing equations is derived and a solution that tracks the boundaries obtained via the use of a numerical ode solver in the cases where the exponent in the diffusivity model is zero linear diffusion or infinite closed from solutions are found for the full range of the diffusivity exponents 0 the similarity solution shows that when r a b 0 4 there is no distinction in the predicted speeds of the moving boundaries further within the range of physically meaningful values of the diffusivity exponent i e 0 2 reasonable agreement in predictions extents up to r a b 0 7 in addition to the similarity solution a fixed grid enthalpy like solution is also proposed predictions obtained with this solution closely match those obtained with the similarity solution

existence of an exact solution for a one phase stefan problem with nonlinear thermal coefficients from tirskii s method abstract the mathematical analysis of a one phase lame clapeyron stefan problem with nonlinear thermal coefficients following g a tirskii two exact solutions of stefan s nonlinear problem sov phys dokl 4 1959 288 292 is obtained two related cases are considered one of them has a temperature condition on the fixed face x 0 and the other one has a flux condition of the type q 0 t q 0 0 we obtain in both cases sufficient conditions for data in order to have the existence of an explicit solution of a similarity type which is given by using a double fixed point

relative oscillation non oscillation criteria for perturbed periodic dirac systems under minimal hypotheses sufficient criteria for a perturbed dirac system to be relatively oscillatory or non oscillatory at with respect to a reference equation with periodic coefficients are proved my means of an asymptotic analysis of generalized prufer angles as illustrated by an example they help decide whether the number of eigenvalues in gaps of the essential spectrum is finite or infinite the critical perturbations naturally occur in the partial wave analysis of spherically symmetric dirac operators

minimax principle for the dirac equation the problem of calculating the eigenvalues of the dirac equation by the finite basis expansion method is studied bounds for the eigenvalues are obtained explaining the numerical results on the spectrum that have been observed previously it is argued that the problem of variational collapse can be avoided by finding the minimum over the wave function large component of the maximum over the wave function small component of the energy functional a numerical example is discussed

on perturbations of quasiperiodic schrodinger operators abstract using relative oscillation theory and the reducibility result of eliasson we study perturbations of quasiperiodic schrodinger operators in particular we derive relative oscillation criteria and eigenvalue asymptotics for critical potentials

on the structure of positive maps finite dimensional case a natural and intrinsic characterization of the structure of the set c of positive unital maps is given i e it is shown that c is isometrically isomorphic to the subset d of bp positive density matrices endowed with the geometry given by the norm dual to the grothendieck projective norm the structure of d is determined by the set of its exposed points and finally a characterization of exposed points of d in terms of convex analysis is presented this seems to be an answer to an old open problem characterization of the structure of the set of positive maps studied both in quantum information and operator algebras our arguments are based on the concept of exposed points and links between tensor products and mapping spaces it should be emphasized that the concept of exposed point depends strongly on the geometry hence the role of the norm is crucial

self adjointness for dirac operators via hardy dirac inequalities distinguished self adjoint extensions of dirac operators are constructed for a class of singular potentials including coulombic ones up to and including the critical case x 1 the method uses hardy dirac inequalities and quadratic form techniques

explicit solution for a stefan problem with variable latent heat and constant heat flux boundary conditions abstract in voller swenson and paola v r voller j b swenson c paola an analytical solution for a stefan problem with variable latent heat int j heat mass transfer 47 2004 5387 5390 and lorenzo trueba and voller j lorenzo trueba v r voller analytical and numerical solution of a generalized stefan problem exhibiting two moving boundaries with application to ocean delta formation j math anal appl 366 2010 538 549 a model associated with the formation of sedimentary ocean deltas is studied through a one phase stefan like problem with variable latent heat motivated by these works we consider a two phase stefan problem with variable latent of fusion and initial temperature and constant heat flux boundary conditions we obtain the sufficient condition on the data in order to have an explicit solution of a similarity type of the corresponding free boundary problem for a semi infinite material moreover the explicit solution given in the first quoted paper can be recovered for a particular case by taking a null heat flux condition at the infinity

non consistent approximations of self adjoint eigenproblems application to the supercell method in this article we introduce a general theoretical framework to analyze non consistent approximations of the discrete eigenmodes of a self adjoint operator we focus in particular on the discrete eigenvalues laying in spectral gaps we first provide a priori error estimates on the eigenvalues and eigenvectors in the absence of spectral pollution we then show that the supercell method for perturbed periodic schrodinger operators falls into the scope of our study we prove that this method is spectral pollution free and we derive optimal convergence rates for the planewave discretization method taking numerical integration errors into account some numerical illustrations are provided

a new approach to spectral approximation a new technique for approximating eigenvalues and eigenvectors of a self adjoint operator is presented the method does not incur spectral pollution uses trial spaces from the form domain has a self adjoint algorithm and exhibits superconvergence

effective prufer angles and relative oscillation criteria abstract we present a streamlined approach to relative oscillation criteria based on effective prufer angles adapted to the use at the edges of the essential spectrum based on this we provided a new scale of oscillation criteria for general sturm liouville operators which answer the question whether a perturbation inserts a finite or an infinite number of eigenvalues into an essential spectral gap as a special case we recover and generalize the gesztesy unal criterion which works below the spectrum and contains classical criteria by kneser hartman hille and weber and the well known results by rofe beketov including the extensions by schmidt

on quantum quadratic operators of bm_2 mathbb c and their dynamics in the present paper we study nonlinear dynamics of quantum quadratic operators q q o acting on the algebra of 2 times 2 matrices bm 2 bc first we describe q q o with haar state as well as quadratic operators with the kadison schwarz property by means of such a description we provide an example of q q o which is not the kadision schwarz operator then we study stability of dynamics of q q o

the spectrum of relativistic one electron atoms according to bethe and salpeter bethe and salpeter introduced a relativistic equation different from the bethe salpeter equation which describes relativistic multi particle systems here we will begin some basic work concerning its mathematical structure in particular we show self adjointness of the one particle operator which will be a consequence of a sharp sobolev type inequality yielding semi boundedness of the corresponding sesquilinear form moreover we locate the essential spectrum of the operator and show the absence of singular continuous spectrum

on the eigenvalues of operators with gaps application to dirac operators this paper is devoted to a general min max characterization of the eigenvalues in a gap of the essential spectrum of a self adjoint unbounded operator we prove an abstract theorem then we apply it to the case of dirac operators with a coulomb like potential the result is optimal for the coulomb potential

cones of positive maps and their duality relations the structure of cones of positive and k positive maps acting on a finite dimensional hilbert space is investigated special emphasis is given to their duality relations to the sets of superpositive and k superpositive maps we characterize k positive and k superpositive maps with regard to their properties under taking compositions a number of results obtained for maps are also rephrased for the corresponding cones of block positive k block positive separable and k entangled operators due to the jamiolkowski choi isomorphism generalizations to a situation where no such simple isomorphism is available are also made employing the idea of mapping cones as a side result to our discussion we show that extreme entanglement witnesses which are optimal should be of special interest in entanglement studies

on approximation of the eigenvalues of perturbed periodic schrodinger operators this paper addresses the problem of computing the eigenvalues lying in the gaps of the essential spectrum of a periodic schrodinger operator perturbed by a fast decreasing potential we use a recently developed technique the so called quadratic projection method in order to achieve convergence free from spectral pollution we describe the theoretical foundations of the method in detail and illustrate its effectiveness by several examples

on a theorem of deift and hempel we provide an alternative proof of the main result of deift and hempel 1 on the existence of eigenvalues ofv dimensional schrodinger operatorsh h0 w in spectral gaps ofh0

zeros of the wronskian and renormalized oscillation theory for general sturm liouville operators with separated boundary conditions we prove the following if e 1 2 r and if u 1 2 solve the differential equation hu j e ju j j 1 2 and respectively satisfy the boundary condition on the left right then the dimension of the spectral projection p e 1 e 2 h of h equals the number of zeros of the wronskian of u 1 and u 2

relative oscillation theory for dirac operators we develop relative oscillation theory for one dimensional dirac operators which rather than measuring the spectrum of one single operator measures the difference between the spectra of two different operators this is done by replacing zeros of solutions of one operator by weighted zeros of wronskians of solutions of two different operators in particular we show that a sturm type comparison theorem still holds in this situation and demonstrate how this can be used to investigate the number of eigenvalues in essential spectral gaps furthermore the connection with krein s spectral shift function is established as an application we extend a result by k m schmidt on the finiteness infiniteness of the number of eigenvalues in essential spectral gaps of perturbed periodic dirac operators

characterization of partial positive transposition states and measures of entanglement a detailed characterization of partial positive transposition ppt states both in the heisenberg and in the schrodinger picture is given measures of entanglement are defined and discussed in details illustrative examples are provided

approximation of the modified error function abstract in this article we obtain explicit approximations of the modified error function introduced in cho and sunderland 1974 as part of a stefan problem with a temperature dependent thermal conductivity this function depends on a parameter which is related to the thermal conductivity in the original phase change process we propose a method to obtain approximations which is based on the assumption that the modified error function admits a power series representation in accurate approximations are obtained through functions involving error and exponential functions only for the special case in which assumes small positive values we show that the modified error function presents some characteristic features of the classical error function such as monotony concavity and boundedness moreover we prove that the modified error function converges to the classical one when goes to zero

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

dense point spectrum for the one dimensional dirac operator with an electrostatic potential for the one dimensional dirac operator examples of electrostatic potentials with decay behaviour arbitrarily close to coulomb decay are constructed for which the operator has a prescribed set of eigenvalues dense in the whole or part of its essential spectrum a simple proof that the essential spectrum of one dimensional dirac operators with electrostatic potentials is never empty is given in the appendix

explicit solutions for a two phase unidimensional lame clapeyron stefan problem with source terms in both phases a two phase stefan problem with heat source terms of a general similarity type in both liquid and solid phases for a semi infinite phase change material is studied we assume the initial temperature is a negative constant and we consider two different boundary conditions at the fixed face x 0 a constant temperature or a heat flux of the form q0 t q0 0 the internal heat source functions are given by gj x t lt j x2ajt j 1 solid phase j 2 liquid phase where j j are functions with appropriate regularity properties is the mass density l is the fusion latent heat by unit of mass aj2 is the diffusion coefficient x is the spatial variable and t is the temporal variable we obtain for both problems explicit solutions with a restriction for data only for the second boundary conditions on x 0 moreover the equivalence of the two free boundary problems is also proved we generalize the solution obtained in j l menaldi d a tarzia generalized lame clapeyron solution for a one phase source stefan problem comput appl math 12 2 1993 123 142 for the one phase stefan problem finally a particular case where j j 1 2 are of exponential type given by j x exp x dj 2 with x and dj r is also studied in details for both boundary temperature conditions at x 0 this type of heat source terms is important through the use of microwave energy following e p scott an analytical solution and sensitivity study of sublimation dehydration within a porous medium with volumetric heating j heat transfer 116 1994 686 693 we obtain a unique solution of the similarity type for any data when a temperature boundary condition at the fixed face x 0 is considered a similar result is obtained for a heat flux condition imposed on x 0 if an inequality for parameter q0 is satisfied

extension of positive maps into b h soient 7b h et 7b k des espaces de hilbert complexes et soit a un sous espace lineaire auto adjoint norme clos d operateurs bornes b 7b k sur 7b k contenant l operateur identite si on note par b a 7b h les applications positives dans l ensemble b a 7b h des applications lineaires bornees de a dans b 7b h on etudie le probleme d extension pour des applications dans b a 7b h aux applications dans b b 7b k 7b h

essential numerical range of an operator with respect to a coercive form and the approximation of its spectrum by the galerkin method we define the notion of essential numerical range of an operator with respect to a coercive sesquilinear form and we prove convergence of the galerkin approximate eigenvalues which lie outside this region

oscillation and non oscillation of euler type half linear differential equations abstract we investigate oscillatory properties of second order euler type half linear differential equations whose coefficients are given by periodic functions and functions having mean values we prove the conditional oscillation of these equations in addition we prove that the known oscillation constants for the corresponding equations with only periodic coefficients do not change in the studied more general case the presented results are new for linear equations as well

on the difference of n positivity and complete positivity in c algebras abstract for two c algebras a and b we prove that every n positive map of a to b is n 1 positive afterwards completely positive if and only if either a or b has only irreducible representations with dimensions less than n similar characterizations are also shown to be valid when both algebras a and b are replaced by the duals of c algebras or the preduals of von neumann algebras together with other applications

an analytical proof of hardy like inequalities related to the dirac operator we prove some sharp hardy type inequalities related to the dirac operator by elementary direct methods some of these inequalities have been obtained previously using spectral information about the dirac coulomb operator our results are stated under optimal conditions on the asymptotics of the potentials near zero and near infinity

merging of positive maps a construction of various classes of positive maps on matrix algebras abstract for two positive maps i b k i b h i i 1 2 we construct a new linear map b h b k where k k 1 k 2 c h h 1 h 2 c by means of some additional ingredients such as operators and functionals we call it a merging of maps 1 and 2 the properties of this construction are discussed in particular conditions for positivity of as well as for 2 positivity complete positivity optimality and indecomposability are provided in particular we show that for a pair composed of 2 positive and 2 copositive maps there is an indecomposable merging of them one of our main results asserts that for a canonical merging of a pair composed of completely positive and completely copositive extremal maps their canonical merging is an exposed positive map this result provides a wide class of new examples of exposed positive maps as an application new examples of entangled ppt states are described

efficient iterative method for solving the dirac kohn sham density functional theory we present for the first time an efficient iterative method to directlysolve the four component dirac kohn sham dks density functional theory due to the existence of the negative energy continuum in the dks operator the existing iterative techniques for solving the kohn sham systems cannot be efficiently applied to solve the dks systems the key component of our method is a novel filtering step f which acts as a preconditioner in the framework of the locally optimal block preconditioned conjugate gradient lobpcg method the resulting method dubbed the lobpcg f method is able to compute the desired eigenvalues and eigenvectors in the positive energy band without computing any state in the negative energy band the lobpcgf method introduces mild extra cost compared to the standard lobpcg method and can be easily implemented we demonstrate our method in the pseudopotential framework with a planewave basis set which naturally satisfies the kinetic balance prescription numerical results for pt2 au2 tlf and bi2se3 indicate that the lobpcg f method is a robust and efficient method for investigating the relativistic effect in systems containing heavy

modified prufer and efgp transforms and deterministic models with dense point spectrum we provide a new proof of the theorem of simon and zhu that in the region e for a e energies d 2 dx 2 cos x 0 1 has lyapunov behavior with a quasi classical formula for the lyapunov exponent we also prove lyapunov behavior for a e e 2 2 for the discrete model with v j 2 e j v n 0 if n 1 4 9 the arguments depend on a direct analysis of the equations for the norm of a solution

approximations of strongly continuous families of unbounded self adjoint operators the problem of approximating the discrete spectra of families of self adjoint operators that are merely strongly continuous is addressed it is well known that the spectrum need not vary continuously as a set under strong perturbations however it is shown that under an additional compactness assumption the spectrum does vary continuously and a family of symmetric finite dimensional approximations is constructed an important feature of these approximations is that they are valid for the entire family uniformly an application of this result to the study of plasma instabilities is illustrated

on a characterization of positive maps drawing on results of choi stormer and woronowicz we present a nearly complete characterization of certain important classes of positive maps in particular we construct a general class of positive linear maps acting between two matrix algebras b h and b k where h and k are finite dimensional hilbert spaces it turns out that elements of this class are characterized by operators from the dual cone of the set of all separable states on b h k subsequently the relation between entanglements and positive maps is described finally a new characterization of the cone b h b k is given

eigenvalue excluding for perturbed periodic one dimensional schrodinger operators subject of investigation in this paper is a one dimensional schrodinger equation where the potential is a sum of a periodic function and a perturbation decaying at graphic 1 it is well known that the essential spectrum consists of spectral bands and that there may or may not be additional eigenvalues below the lowest band or in the gaps between the bands while enclosures for gap eigenvalues can comparatively easily be obtained from numerical approximations e g by d weinstein s bounds there seems to be no method available so far which is able to exclude eigenvalues in spectral gaps i e which identifies subregions of a gap which contain no eigenvalues here we propose such a method it makes heavy use of computer assistance nevertheless the results are completely rigorous in the strict mathematical sense because all computational errors are taken into account 1 embed inline graphic 1 gif

existence of the exponentially localised wannier functions a partial answer theorem 1 below to a problem concerning analytic and periodic families of projections in hilbert spaces is given as a consequence the existence of exponentially localised wannier functions corresponding to nondegenerated bands of arbitrary three dimensional crystals is proved

modified prufer angle and conditional oscillation of perturbed linear and half linear differential equations abstract the research and results described in this paper belong to the qualitative theory of differential equations more precisely the partial differential equations with the one dimensional p laplacian using a method whose core is formed by the prufer technique we identify a borderline case between oscillatory and non oscillatory equations moreover we are able to decide whether the studied equations are oscillatory or not even in the so called critical i e the borderline case the advantage of our approach is the fact that we obtain new and strong results for linear and half linear equations i e the equations with the one dimensional p laplacian at the same time in addition we are able to work with equations whose coefficients are non constant and non periodic the novelty of our results is documented by examples and corollaries

oscillation and non oscillation criteria for linear and half linear difference equations abstract we find very general classes of linear and half linear difference equations which are conditionally oscillatory we identify the critical oscillation constant whose value implies the oscillation or non oscillation of studied equations our results are divided into oscillatory and non oscillatory theorems which determine the critical oscillation constant for coefficients given by sequences having mean values in addition our approach enables to analyse also the oscillatory properties of equations whose coefficients are not given by sequences with mean values we point out that the obtained results are new even for linear equations with periodic coefficients such consequences are formulated at the end of this paper

critical coupling constants and eigenvalue asymptotics of perturbed periodic sturm liouville operators perturbations of asymptotic decay c r 2 arise in the partial wave analysis of rotationally symmetric partial differential operators we show that for each end point 0 of the spectral bands of a perturbed periodic sturm liouville operator there is a critical coupling constant c crit such that eigenvalues in the spectral gap accumulate at 0 if and only if c c crit 1 the oscillation theoretic method used in the proof also yields the asymptotic distribution of the eigenvalues near 0

positive maps positive polynomials and entanglement witnesses we link the study of positive quantum maps block positive operators and entanglement witnesses with problems related to multivariate polynomials for instance we show how indecomposable block positive operators relate to biquadratic forms that are not sums of squares although the general problem of describing the set of positive maps remains open in some particular cases we solve the corresponding polynomial inequalities and obtain explicit conditions for positivity

a class of positive atomic maps we construct a new class of positive indecomposable maps in the algebra of d d complex matrices these maps are characterized by the weakest positivity property and for this reason they are called atomic this class provides a new rich family of atomic entanglement witnesses which define an important tool for investigating quantum entanglement it turns out that they are able to detect states with the weakest quantum entanglement

on quantum quadratic operators of m2 c and their dynamics abstract in the present paper we study nonlinear dynamics of quantum quadratic operators q q o acting on the algebra of 2 2 matrices m 2 c first we describe q q o with haar state as well as quadratic operators with the kadison schwarz property by means of such a description we provide an example of q q o which is not the kadison schwarz operator then we study stability of dynamics of q q o

spectral pollution and how to avoid it this paper devoted to the study of spectral pollution contains both abstract results and applications to some self adjoint operators with a gap in their essential spectrum occuring in quantum mechanics first we consider galerkin basis which respect the decomposition of the ambient hilbert space into a direct sum h ph oplus 1 p h given by a fixed orthogonal projector p and we localize the polluted spectrum exactly this is followed by applications to periodic schrodinger operators pollution is absent in a wannier type basis and to dirac operator several natural decompositions are considered in the second part we add the constraint that within the galerkin basis there is a certain relation between vectors in ph and vectors in 1 p h abstract results are proved and applied to several practical methods like the famous kinetic balance of relativistic quantum mechanics

radon nikodym derivatives of quantum operations given a completely positive cp map t there is a theorem of the radon nikodym type w b arveson acta math 123 141 1969 v p belavkin and p staszewski rep math phys 24 49 1986 that completely characterizes all cp maps s such that t s is also a cp map this theorem is reviewed and several alternative formulations are given along the way we then use the radon nikodym formalism to study the structure of order intervals of quantum operations as well as a certain one to one correspondence between cp maps and positive operators already fruitfully exploited in many quantum information theoretic treatments we also comment on how the radon nikodym theorem can be used to derive norm estimates for differences of cp maps in general and of quantum operations in particular

eigenvalues in gaps of perturbed periodic dirac operators numerical evidence this paper presents a method for the numerical investigation of the distribution of the eigenvalues introduced into a spectral gap of a periodic dirac system by a perturbation of the type of the angular momentum term a number of examples illustrate the effectiveness of the method and show the remarkable accuracy of the strong coupling asymptotic formula even for small values of the perturbation coupling constant furthermore the results shed some light on the spectrum in the exceptional gap of radially periodic three dimensional dirac operators

on the approximation of spectra of linear operators on hilbert spaces abstract we present several new techniques for approximating spectra of linear operators not necessarily bounded on an infinite dimensional separable hilbert space our approach is to take well known techniques from finite dimensional matrix analysis and show how they can be generalized to an infinite dimensional setting to provide approximations of spectra of elements in a large class of operators we conclude by proposing a solution to the general problem of approximating the spectrum of an arbitrary bounded operator by introducing the n pseudospectrum and argue how that can be used as an approximation to the spectrum

critical oscillation constant for euler type dynamic equations on time scales in this paper we study the second order dynamic equation on the time scale t of the form r t y d email protected t email protected t y s 0 wherer qare positive rd continuous periodic functions with inf r t email protected t 0 and c is an arbitrary real constant this equation corresponds to euler type differential resp euler type difference equation for continuous resp discrete case our aim is to prove that this equation is conditionally oscillatory i e there exists a constant c 0 such that studied equation is oscillatory for c c and non oscillatory for c c

relative oscillation theory weighted zeros of the wronskian and the spectral shift function we develop an analog of classical oscillation theory for sturm liouville operators which rather than measuring the spectrum of one single operator measures the difference between the spectra of two different operators

edge states in ordinary differential equations for dislocations in this article we study schrodinger operators on the real line when the external potential represents a dislocation in a periodic medium we study how the spectrum varies with the dislocation parameter we introduce several integer valued indices including the chern number for bulk indices and various spectral flows for edge indices we prove that all these indices coincide providing a proof of a bulk edge correspondence in this case the study is also made for dislocations in dirac models on the real line we prove that 0 is always an eigenvalue of such operators

a family of indecomposable positive linear maps based on entangled quantum states abstract we introduce a new family of indecomposable positive linear maps based on entangled quantum states central to our construction is the notion of an unextendible product basis the construction lets us create indecomposable positive linear maps in matrix algebras of arbitrary high dimension

on the structure of positive maps ii low dimensional matrix algebras we use a new idea that emerged in the examination of exposed positive maps between matrix algebras to investigate in more detail the differences and similarities between unital positive maps on m2 c and m3 c our main tool stems from classical grothendieck theorem on tensor product of banach spaces and is an older and more general version of choi jamiolkowski isomorphism between positive maps and block positive choi matrices it takes into account the correct topology on the latter set that is induced by the uniform topology on positive maps in this setting we show that in m2 c case a large class of nice positive maps can be generated from the small set of maps represented by self adjoint unitaries 2px with x maximally entangled vector and p 1 with p rank 1 projector we indicate problems with passing this result to m3 c case among similarities in both m2 c and m3 c cases any unital positive map represented by self adjoint unitary is unitarily equivalent to the transposition map consequently

on the existence of eigenvalues of the schrodinger operatorh λw in a gap of σ h the authors prove a number of results asserting the existence of eigenvalues of the schrodinger operatorh w in a gap of h whereh v andv andw are bounded the existence of these eigenvalues is an important element in the theory of the colour of crystals the basic theorems are proved in v stronger results forv 1 are also presented

modified prufer and efgp transforms and the spectral analysis of one dimensional schrodinger operators using control of the growth of the transfer matrices wediscuss the spectral analysis of continuum and discrete half line schrodinger operators with slowly decaying potentials among our results we show if where w has compact support and then h has purely a c resp purely s c spectrum on 0 if for n 1 2 a n potentials where a n are independent identically distributed random variables with e a n 0 e a 2 n 1 and 2 we find singular continuous spectrum with explicitly computable fractional hausdorff dimension

on the convergence of second order spectra and multiplicity the notion of second order relative spectrum of a self adjoint operator acting on a hilbert space has been studied recently in connection with the phenomenon of spectral pollution in the galerkin method in this paper we examine how the second order spectrum encodes precise information about the multiplicity of the isolated eigenvalues of the underlying operator our theoretical findings are supported by various numerical experiments on the computation of guaranteed eigenvalue inclusions via finite element bases

generalized prufer variables for perturbations of jacobi and cmv matrices abstract prufer variables are a standard tool in spectral theory developed originally for perturbations of the free schrodinger operator we adapt the generalized prufer variables introduced by kiselev remling simon for perturbations of an arbitrary schrodinger operator to the setting of jacobi and szeg recursions we present an application to random l 2 perturbations of jacobi and cmv matrices and an application to decaying oscillatory perturbations of periodic jacobi and cmv matrices

rofe beketov s formula on time scales we establish the rofe beketov formula for the second order sturm liouville dynamic equation on time scales this formula shows how to express a second linearly independent solution of this equation if one solution is known in contrast to the d alembert formula sometimes also called the reduction of order formula no restriction on the oscillatory behavior of the known solution is needed