a failing eigenfunction expansion associated with an indefinite sturm liouville problem we consider the indefinite sturm liouville problem f rf f 1 f 1 0 where r l1 1 1 satisfies xr x 0 conditions are presented such that the normed eigenfunctions fn form a riesz basis of the hilbert space l r 2 1 1 using known results for a modified problem the main focus is on the non riesz basis case we construct a function f l r 2 1 1 having no eigenfunction expansion f nfn furthermore a sequence n l2 is constructed such that the fourier series nfn does not converge in l r 2 1 1 these problems are closely related to the regularity property of the closed non semibounded symmetric sesquilinear form t u v u v pdx with dirichlet boundary conditions in l2 1 1 where p 1 r for the associated operator tt we construct elements in the difference between domt and the domain of the associated regular closed form i e dom tt 1 2

bulk universality for generalized wigner matrices consider n n hermitian or symmetric random matrices h where the distribution of the i j matrix element is given by a probability measure ij with a subexponential decay let sigma ij 2 be the variance for the probability measure ij with the normalization property that sum i sigma 2 ij 1 for all j under essentially the only condition that c le n sigma ij 2 le c 1 for some constant c 0 we prove that in the limit n the eigenvalue spacing statistics of h in the bulk of the spectrum coincide with those of the gaussian unitary or orthogonal ensemble gue or goe we also show that for band matrices with bandwidth m the local semicircle law holds to the energy scale m 1

the bessel differential equation and the hankel transform this paper studies the classical second order bessel differential equation in liouville form here the parameter represents the order of the associated bessel functions and is the complex spectral parameter involved in considering properties of the equation in the hilbert function space l2 0 properties of the equation are considered when the order e 0 1 in this case the singular end point 0 is in the limit circle non oscillatory classification in the space l2 0 the equation is in the strong limit point and dirichlet condition at the end point applying the generalised initial value theorem at the singular end point 0 allows of the definition of a single titchmarsh weyl m coefficient for the whole interval 0 in turn this information yields a proof of the hankel transform as an eigenfunction expansion for the case when e 0 1 a result which is not available in the existing literature the application of the principal solution from the end point 0 of the bessel equation as a boundary condition function yields the friedrichs self adjoint extension in l2 0 the domain of this extension has many special known properties of which new proofs are presented

quantization of models with non compact quantum group symmetry modular xxz magnet and lattice sinh gordon model we define and study certain integrable lattice models with non compact quantum group symmetry the modular double of including an integrable lattice regularization of the sinh gordon model and a non compact version of the xxz model their fundamental r matrices are constructed in terms of the non compact quantum dilogarithm our choice of the quantum group representations naturally ensures self adjointness of the hamiltonian and the higher integrals of motion these models are studied with the help of the separation of variables method we show that the spectral problem for the integrals of motion can be reformulated as the problem to determine a subset among the solutions to certain finite difference equations baxter equation and quantum wronskian equation which is characterized by suitable analytic and asymptotic properties a key technical tool is the so called operator for which we give an explicit construction our results allow us to establish some connections to related results and conjectures on the sinh gordon theory in continuous spacetime our approach also sheds some light on the relations between massive and massless models in particular the sinh gordon and liouville theories from the point of view of their integrable structures

level statistics and localization transitions of levy matrices this work provides a thorough study of levy or heavy tailed random matrices lms by analyzing the self consistent equation on the probability distribution of the diagonal elements of the resolvent we establish the equation determining the localization transition and obtain the phase diagram using arguments based on supersymmetric field theory and dyson brownian motion we show that the eigenvalue statistics is the same one as of the gaussian orthogonal ensemble in the whole delocalized phase and is poisson like in the localized phase our numerics confirm these findings valid in the limit of infinitely large lms but also reveal that the characteristic scale governing finite size effects diverges much faster than a power law approaching the transition and is already very large far from it this leads to a very wide crossover region in which the system looks as if it were in a mixed phase our results together with the ones obtained previously now provide a complete theory of levy matrices

on quantum separation of variables we present a new approach to construct the separate variables basis leading to the full characterization of the transfer matrix spectrum of quantum integrable lattice models the basis is generated by the repeated action of the transfer matrix itself on a generically chosen state of the hilbert space the fusion relations for the transfer matrix stemming from the yang baxter algebra properties provide the necessary closure relations to define the action of the transfer matrix on such a basis in terms of elementary local shifts leading to a separate transfer matrix spectral problem hence our scheme extends to the quantum case a key feature of the liouville arnold classical integrability framework where the complete set of conserved charges defines both the level manifold and the flows on it leading to the construction of action angle variables we work in the framework of the quantum inverse scattering method as a first example of our approach we give the construction of such a basis for models associated with y gln and argue how it extends to their trigonometric and elliptic versions then we show how our general scheme applies concretely to fundamental models associated with the y gl2 and y gl3 r matrices leading to the full characterization of their spectrum for y gl2 and its trigonometric deformation a particular case of our method reproduces sklyanin s construction of separate variables for y gl3 it gives new results in particular through the proper identification of the shifts acting on the separate basis we stress that our method also leads to the full characterization of the spectrum of other known quantum integrable lattice models including in particular trigonometric and elliptic spin chains open chains with general integrable boundaries and further higher rank cases that we will describe in forthcoming publications

sov approach for integrable quantum models associated with general representations on spin 1 2 chains of the 8 vertex reflection algebra the analysis of the transfer matrices associated with the most general representations of the 8 vertex reflection algebra on spin 1 2 chains is here implemented by introducing a quantum separation of variables sov method which generalizes to these integrable quantum models the method first introduced by sklyanin for representations reproducing in their homogeneous limits the open xyz spin 1 2 quantum chains with the most general integrable boundary conditions we explicitly construct representations of the 8 vertex reflection algebras for which the transfer matrix spectral problem is separated then in these sov representations we get the complete characterization of the transfer matrix spectrum eigenvalues and eigenstates and its non degeneracy moreover we present the first fundamental step toward the characterization of the dynamics of these models by deriving determinant formulae for the matrix elements of the identity on separated states which particularly apply to transfer matrix eigenstates a comparison of our analysis of the 8 vertex reflection algebra with that of niccoli g 2012 j stat mech p10025 faldella s et al 2014 j stat mech p01011 for the 6 vertex leads to an interesting remark in that there is a profound similarity in both the characterization of the spectral problems and the scalar products which exists for these two different realizations of the reflection algebra once they are described by the sov method as will be shown in a future publication this remarkable similarity will be the basis of a simultaneous determination of the form factors of local operators of integrable quantum models associated with general reflection algebra representations of both 8 vertex and 6 vertex type

the eigenvector moment flow and local quantum unique ergodicity we prove that the distribution of eigenvectors of generalized wigner matrices is universal both in the bulk and at the edge this includes a probabilistic version of local quantum unique ergodicity and asymptotic normality of the eigenvector entries the proof relies on analyzing the eigenvector flow under the dyson brownian motion the key new ideas are 1 the introduction of the eigenvector moment flow a multi particle random walk in a random environment 2 an effective estimate on the regularity of this flow based on maximum principle and 3 optimal finite speed of propagation holds for the eigenvector moment flow with very high probability

theory of levy matrices we investigate the statistical properties of the spectrum of large symmetrical matrices with each element mathit h mathit i mathit j chosen according to a broad distribution ensuremath rho h decaying for large h as mathit h mathrm ensuremath 1 mathrm ensuremath mathrm ensuremath mu for ensuremath mu g2 mathit h 2 is finite and the well known gaussian orthogonal ensemble goe results are recovered when ensuremath mu 2 the semicircular law is replaced by a density which extends over the whole energy axis furthermore while all states are extended in the case of goe matrices we show numerically and analytically that two mobility edges appear separating extended from localized states with an intermediate mixed phase in between the unusual nature of these localized states is discussed

confluent cauchy and cauchy vandermonde matrices abstract the s c confluent cauchy and confluent cauchy vandermonde matrices are introduced corresponding to an interpolation problem with multiple interpolation nodes inversion formulas are derived and a factorization of confluent loewner matrices by means of confluent cauchy vandermonde matrices is described

a general help inequality connected with symmetric operators in this paper a general help hardy everitt littlewood polya inequality is considered which is connected with a symmetric operator in a hilbert space and abstract boundary mappings a criterion for the validity of such an inequality in terms of the abstract titchmarsh weyl function is proved and applied to sturm liouville operators difference operators a hamiltonian system and a block operator matrix

local spectral statistics of gaussian matrices with correlated entries we prove optimal local law bulk universality and non trivial decay for the off diagonal elements of the resolvent for a class of translation invariant gaussian random matrix ensembles with correlated entries

structure of matrix elements in the quantum toda chain we consider the quantum toda chain using the method of separation of variables we show that the matrix elements of operators in the model are written in terms of a finite number of deformed abelian integrals the properties of these integrals are discussed we explain that these properties are necessary in order to provide the correct number of independent operators a comparison with the classical theory is made

help inequalities for limit circle and regular problems everitt s criterion for the validity of the generalized hardy littlewood inequality presupposes that the associated differential equation is singular at one end point of the interval of definition and is in the strong limit point case at the end point in this paper we investigate the cases when the differential equation is in the limit circle case and non oscillatory at the singular end point and when both end points of the interval are regular

optimal bounds for convergence of expected spectral distributions to the semi circular law let mathbf x x jk j k 1 n denote a hermitian random matrix with entries x jk which are independent for 1 le j le k le n we consider the rate of convergence of the empirical spectral distribution function of the matrix mathbf x to the semi circular law assuming that mathbf e x jk 0 mathbf e x jk 2 1 and that begin aligned sup n ge 1 sup 1 le j k le n mathbf e x jk 4 mu 4 infty end aligned and begin aligned sup 1 le j k le n x jk le d 0n frac 1 4 end aligned by means of a recursion argument it is shown that the kolmogorov distance between the expected spectral distribution of the wigner matrix mathbf w frac 1 sqrt n mathbf x and the semicircular law is of order o n 1

fixed energy universality for generalized wigner matrices we prove the wigner dyson mehta conjecture at fixed energy in the bulk of the spectrum for generalized symmetric and hermitian wigner matrices previous results concerning the universality of random matrices either require an averaging in the energy parameter or they hold only for hermitian matrices if the energy parameter is fixed we develop a homogenization theory of the dyson brownian motion and show that microscopic universality follows from mesoscopic statistics 2015 wiley periodicals inc

the baxter bazhanov stroganov model separation of variables and the baxter equation the baxter bazhanov stroganov model also known as the 2 model has attracted much interest because it provides a tool for solving the integrable chiral potts model it can be formulated as a face spin model or via cyclic l operators using the latter formulation and the sklyanin kharchev lebedev approach we give the explicit derivation of the eigenvectors of the component bn of the monodromy matrix for the fully inhomogeneous chain of finite length for the periodic chain we obtain the baxter t q equations via separation of variables the functional relations for the transfer matrices of the 2 model guarantee nontrivial solutions to the baxter equations for the n 2 case which is the free fermion point of a generalized ising model the baxter equations are solved explicitly

on a series inequality for limit circle and finite problems brown and evans proposed a criterion for the validity of their series analogue of everitt s extension of the hardy littlewood inequality presupposes that the associated difference equation is in the strong limit point at infinity in this paper we investigate the case when the difference equation involved is in the limit circle case and non oscillatory and also when the interval is finite

bulk universality for generalized wigner matrices with few moments in this paper we consider n times n real generalized wigner matrices whose entries are only assumed to have finite 2 varepsilon th moment for some fixed but arbitrarily small varepsilon 0 we show that the stieltjes transforms m n z of these matrices satisfy a weak local semicircle law on the nearly smallest possible scale when eta mathfrak i z is almost of order n 1 as a consequence we establish bulk universality for local spectral statistics of these matrices at fixed energy levels both in terms of eigenvalue gap distributions and correlation functions meaning that these statistics converge to those of the gaussian orthogonal ensemble in the large n limit

the open xxx spin chain in the sov framework scalar product of separate states in our previous paper kitanine et al 2017 j phys a math theor 50 224001 we have obtained for the xxx spin 1 2 heisenberg open chain new determinant representations for the scalar products of separate states in the quantum separation of variables sov framework in this article we perform a similar study in a more complicated case the xxz open spin 1 2 chain with the most general integrable boundary terms we first revisit the sov solution of this model using a matrix version of the vertex irf gauge transformation so as to reduce one of the boundary k matrices to a diagonal form as usual within the sov approach the scalar products of separate states are computed in terms of dressed vandermonde determinants having an intricate dependency on the inhomogeneity parameters we show that these determinants can be transformed into different ones in which the homogeneous limit can be taken straightforwardly these representations generalize in a non trivial manner to the trigonometric case the expressions found previously in the rational case we also show that generically all scalar products can be expressed in a form which is similar to although more cumbersome than the well known slavnov determinant representation for the scalar products of the bethe states of the periodic chain considering a special choice of the boundary parameters relevant in the thermodynamic limit to describe the half infinite chain with a general boundary we particularize these representations to the case of one of the two states being an eigenstate we obtain simplified formulas that should be of direct use to compute the form factors and correlation functions of this model

the similarity problem for indefinite sturm liouville operators and the help inequality abstract we study two problems the first one is the similarity problem for the indefinite sturm liouville operator a sgn x d w d x d r d x acting in l w 2 b b it is assumed that w r l loc 1 b b are even and positive a e on b b the second object is the so called help inequality 0 b 1 r f d x 2 k 2 0 b f 2 w d x 0 b 1 w 1 r f 2 w d x where the coefficients w r l loc 1 0 b are positive a e on 0 b both problems are well understood when the corresponding sturm liouville differential expression is regular the main objective of the present paper is to give criteria for both the validity of the help inequality and the similarity to a self adjoint operator in the singular case namely we establish new criteria formulated in terms of the behavior of the corresponding weyl titchmarsh m functions at 0 and at as a biproduct of this result we show that both problems are closely connected namely the operator a is similar to a self adjoint one precisely if the help inequality with w r and r w is valid next we characterize the behavior of m functions in terms of coefficients and then these results enable us to reformulate the obtained criteria in terms of coefficients finally we apply these results for the study of the two way diffusion equation also known as the time independent fokker planck equation

a return to the hardy littlewood integral inequality this paper is concerned with the extension and generalization of a classic integral inequality first considered by hardy and littlewood in 1932 the integral inequality is big int b a p f 2 q f 2 big 2 leqslant k int b a w f 2 int b a w w 1 pf qf 2 f in delta where k is a positive number the coefficients p q and w are real valued and w 0 on the interval a b infty a b leqslant infty and p 1 q w in l loc a b delta is the maximal linear manifold of functions f for which the two integrals on the right hand side are both finite the hardy littlewood inequality is the special case a 0 b infty p w 1 and q 0 on 0 infty here k 4 and all the cases of equality can be identified in this paper the results of everitt in 1971 connecting the above inequality with the titchmarsh weyl m coefficient of the associated differential equation py qy lambda wy on a b are restated in an improved form and are shown to hold under the above minimal conditions on the coefficients p q and w it is proved that the best possible value for the number k satisfies 1 k leqslant infty and that within these bounds all values of k can be realized in the course of establishing these results new information of independent interest is obtained about the behaviour of the m coefficient near the real axis examples are given to exhibit the wide range of possibilities for the integral inequality recent results on the asymptotic form of the m coefficient and on the transformation theory of the associated differential equation are shown to have applications for the general properties of the inequality all recent contributions to this general integral inequality are listed in the references and their content is surveyed during the course of the paper

separation of variables in the classical integrable sl 3 magnetic chain there are two fundamental problems studied by the theory of hamiltonian integrable systems integration of equations of motion and construction of action angle variables a third problem however should be added to the list separation of variables though much simpler thant the two others it has important relations to quantum integrability separation of variables is constructed for thesl 3 magnetic chain an example of an integrable model associated to a nonhyperelliptic algebraic curve

bulk universality of sparse random matrices we consider the adjacency matrix of the ensemble of erd s renyi random graphs which consists of graphs on n vertices in which each edge occurs independently with probability p we prove that in the regime pn 1 these matrices exhibit bulk universality in the sense that both the averaged n point correlation functions and distribution of a single eigenvalue gap coincide with those of the goe our methods extend to a class of random matrices which includes sparse ensembles whose entries have different variances

the b and m shapiro conjecture in real algebraic geometry and the bethe ansatz we prove the b and m shapiro conjecture that if the wronskian of a set of polynomials has real roots only then the complex span of this set of polynomials has a basis consisting of polynomials with real coefficients this in particular implies the following result if all ramification points of a parametrized rational curve 1 r lie on a circle in the riemann sphere 1 then maps this circle into a suitable real subspace r r the proof is based on the bethe ansatz method in the gaudin model the key observation is that a symmetric linear operator on a euclidean space has real spectrum in appendix a we discuss properties of differential operators associated with bethe vectors in the gaudin model in particular we prove a statement which may be useful in complex algebraic geometry it claims that certain schubert cycles in a grassmannian intersect transversally if the spectrum of the corresponding gaudin hamiltonians is simple in appendix b we formulate a conjecture on reality of orbits of critical points of master functions and prove this conjecture for master functions associated with lie algebras of types a r b r and c r

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

bulk eigenvalue statistics for random regular graphs ak was partly supported by swiss national science foundation grant 144662 hty was partly supported by nsf grant dms 1307444 and a simons investigator fellowship

characterizations of monotone o regularly varying functions by means of indefinite eigenvalue problems and help type inequalities abstract connections between different areas of mathematical analysis are obtained regular variation indefinite operator theory and help type inequalities using a result by parfyonov the nondecreasing and so called positively increasing functions induce precisely those indefinite kre n feller eigenvalue problems such that the eigenfunctions have the riesz basis property furthermore these functions induce precisely those lebesgue stieltjes measures such that the associated help type inequalities are valid the so called dual kre n feller eigenvalue problems allow a similar characterization of the class of nondecreasing o regularly varying functions

a general version of the hardy littlewood polya everitt help inequality the inequality 0 1 below is naturally associated with the equation pu qu u by assuming that one end point of the interval a b is regular and the other limit point for this equation everitt characterized the best constant k in tems of spectral properties of the equation this paper sketches a theory for more general inequalities 0 2 0 3 similarly related to the equation su tu here s and t are ordinary symmetric differential expressions a characterization of the best constants in 0 2 0 3 is given which generalises that of everitt for the case when s is of order 1 and t is multiplication by a positive function all possible inequalities are given together with the best constants and cases of equality furthermore an example is given of a valid inequality 0 1 on an interval with both end points regular for the corresponding differential equation this contradicts a conjecture by everitt and evans finally the general theory for the left definite inequality 0 3 is specialised to the case when s is a sturm liouville expression a family of examples is given for which the best constants can be explicitly calculated

form factors and complete spectrum of xxx antiperiodic higher spin chains by quantum separation of variables the antiperiodic transfer matrices associated to higher spin representations of the rational 6 vertex yang baxter algebra are analyzed by generalizing the approach introduced recently in the framework of sklyanin s quantum separation of variables sov for cyclic representations spin 1 2 highest weight representations and also for spin 1 2 representations of the 6 vertex reflection algebra such sov approach allow us to derive exactly results which represent complicate tasks for more traditional methods based on bethe ansatz and baxter q operator in particular we both prove the completeness of the sov characterization of the transfer matrix spectrum and its simplicity then the derived characterization of local operators by sklyanin s quantum separate variables and the expression of the scalar products of separate states by determinant formulae allow us to compute the form factors of the local spin operators by one determinant formulae similar to those of the scalar products

an antiperiodic dynamical six vertex model i complete spectrum by sov matrix elements of the identity on separate states and connections to the periodic eight vertex model the spin 1 2 highest weight representations of the dynamical six vertex and the standard eight vertex yang baxter algebra on a finite chain are considered in this paper in particular the integrable quantum models associated with the corresponding transfer matrices under antiperiodic boundary conditions for the dynamical six vertex case and periodic boundary conditions for the eightvertexcaseareanalyzedhere fortheantiperiodicdynamicalsix vertextransfer matrix defined on chains with an odd number of sites we adapt sklyanin s quantum separation of variable sov method and explicitly construct the sov representations from the original space of the representations in this way we provide the complete characterization of the eigenvalues and the eigenstates proving also the simplicity of its spectrum moreover we characterize the matrix elements of the identity on separated states of this model by determinant formulae the matrices entering these determinants have elements given by sums over the sov spectrum of the product of the coefficients of the separate states this sov analysis is done without any need to be reduced to the case of the so called elliptic roots of unit and the results derived here define the required setup to extend to the dynamical six vertex model the approach recently developed by the author and collaborators to compute the form factors of the local operators in the sov framework for the periodic eight vertex transfer matrix we prove that its eigenvalues have to satisfy a fixed system of equations in the case of a chain with an odd number of sites this system of equations is the same entering in the sov characterization of the antiperiodic dynamical six vertex transfer matrix spectrum this implies that the set of the periodic eight vertex eigenvalues is contained in the set of the antiperiodic dynamical six vertex eigenvalues a criterion is introduced to find simultaneous eigenvalues of these two transfer matrices and associate with any of such eigenvalues one nonzero eigenstate of the periodic eight vertex transfer matrix by using the sov results moreover a preliminary discussion

the spectrum of heavy tailed random matrices let x n be an n n random symmetric matrix with independent equidistributed entries if the law p of the entries has a finite second moment it was shown by wigner 14 that the empirical distribution of the eigenvalues of x n once renormalized by sqrt n converges almost surely and in expectation to the so called semicircular distribution as n goes to infinity in this paper we study the same question when p is in the domain of attraction of an stable law we prove that if we renormalize the eigenvalues by a constant a n of order n frac 1 alpha the corresponding spectral distribution converges in expectation towards a law mu alpha which only depends on we characterize mu alpha and study some of its properties it is a heavy tailed probability measure which is absolutely continuous with respect to lebesgue measure except possibly on a compact set of capacity zero

on determinant representations of scalar products and form factors in the sov approach the xxx case in the present article we study the form factors of quantum integrable lattice models solvable by the separation of variables sovs method it was recently shown that these models admit universal determinant representations for the scalar products of the so called separate states a class which includes in particular all the eigenstates of the transfer matrix these results permit to obtain simple expressions for the matrix elements of local operators form factors however these representations have been obtained up to now only for the completely inhomogeneous versions of the lattice models considered in this article we give a simple algebraic procedure to rewrite the scalar products and hence the form factors for the sov related models as izergin or slavnov type determinants this new form leads to simple expressions for the form factors in the homogeneous and thermodynamic limits to make the presentation of our method clear we have chosen to explain it first for the simple case of the xxx heisenberg chain with anti periodic boundary conditions we would nevertheless like to stress that the approach presented in this article applies as well to a wide range of models solved in the sov framework

spectral statistics of erdős renyi graphs ii eigenvalue spacing and the extreme eigenvalues we consider the ensemble of adjacency matrices of erd s renyi random graphs i e graphs on n vertices where every edge is chosen independently and with probability p p n we rescale the matrix so that its bulk eigenvalues are of order one under the assumption p n gg n 2 3 we prove the universality of eigenvalue distributions both in the bulk and at the edge of the spectrum more precisely we prove 1 that the eigenvalue spacing of the erd s renyi graph in the bulk of the spectrum has the same distribution as that of the gaussian orthogonal ensemble and 2 that the second largest eigenvalue of the erd s renyi graph has the same distribution as the largest eigenvalue of the gaussian orthogonal ensemble as an application of our method we prove the bulk universality of generalized wigner matrices under the assumption that the matrix entries have at least 4 e moments

central limit theorems for linear statistics of heavy tailed random matrices we show central limit theorems clt for the stieltjes transforms or more general analytic functions of symmetric matrices with independent heavy tailed entries including entries in the domain of attraction of alpha stable laws and entries with moments exploding with the dimension as in the adjacency matrices of erdos renyi graphs for the second model we also prove a central limit theorem of the moments of its empirical eigenvalues distribution the limit laws are gaussian but unlike to the case of standard wigner matrices the normalization is the one of the classical clt for independent random variables

quantization of solitons and the restricted sine gordon model we show how to compute form factors matrix elements of local fields in the restricted sine gordon model at the reflectionless points by quantizing solitons we introduce quantum separated variables in which the hamiltonians are expressed in terms of quantum functions we explicitly describe the soliton wave functions and we explain how the restriction is related to an unusual hermitian structure we also present a semi classical analysis which enlightens the fact that the restricted sine gordon model corresponds to an analytical continuation of the sine gordon model intermediate between sine gordon and kdv

numerical determination of the titchmarsh weyl m coefficient and its applications to help inequalities this paper is concerned with numerical methods for finding m lambda the titchmarsh weyl m coefficient for the singular eigenvalue equation y qy lambda y on 0 infty and the results are applied to the problem of finding best constants for everitt s extension to the hardy little wood polya help integral inequality

a numerical method for the determination of the titchmarsh weyl m coefficient a numerical method for determining the titchmarsh weyl m function for the singular eigenvalue equation py9 9 qy wy on a where a is finite is presented the algorithm based on weyl s theory utilizes a result first used by atkinson to map a point on the real line onto the weyl circle in the complex plane in the limit point case these circles nest and tend to the limit point m using weyl s result for the diameter of the circles error estimates for m are obtained in 1971 w n everitt obtained an extension of an integral inequality of hardy littlewood namely the help inequality he showed that the existence of that inequality is determined by the properties of the null set of im 2 m in view of the major difficulties in analysing m even in the rare cases when it is given explicitly very few examples of the help inequality are known the computational techniques discussed in this paper have been applied to the problem of finding best constants in these inequalities

random covariance matrices universality of local statistics of eigenvalues we study the eigenvalues of the covariance matrix 1 n m m of a large rectangular matrix m mn p ij 1 i p 1 j n whose entries are i i d random variables of mean zero variance one and having finite c0th moment for some sufficiently large constant c0 the main result of this paper is a four moment theorem for i i d covariance matrices analogous to the four moment theorem for wigner matrices established by the authors in acta math 2011 random matrices universality of local eigenvalue statistics see also comm math phys 298 2010 549 572 we can use this theorem together with existing results to establish universality of local statistics of eigenvalues under mild conditions as a byproduct of our arguments we also extend our previous results on random hermitian matrices to the case in which the entries have finite c0th moment rather than exponential decay

localization and delocalization of eigenvectors for heavy tailed random matrices consider an n times n hermitian random matrix with above the diagonal independent entries with alpha stable symmetric distribution and 0 2 we give vanishing bounds on the l p norm of the eigenvectors normalized to have unit l 2 norm on the contrary when 0 alpha 2 3 we prove that these eigenvectors are localized

universality of the local spacing distribution in certain ensembles of hermitian wigner matrices consider an n n hermitian random matrix with independent entries not necessarily gaussian a so called wigner matrix it has been conjectured that the local spacing distribution i e the distribution of the distance between nearest neighbour eigenvalues in some part of the spectrum is in the limit as n the same as that of hermitian random matrices from gue we prove this conjecture for a certain subclass of hermitian wigner matrices