lieb thirring type inequalities and gagliardo nirenberg inequalities for systems this paper is devoted to inequalities of lieb thirring type let v be a nonnegative potential such that the corresponding schrodinger operator has an unbounded sequence of eigenvalues i v i n we prove that there exists a positive constant c such that if d 2 then x i n i v c z rd v d 2 dx

variational estimates for discrete schrodinger operators with potentials of indefinite sign let h be a one dimensional discrete schrodinger operator we prove that if ess h 2 2 then h h 0 is compact and ess h 2 2 we also prove that if h 0 1 4 v 2 has at least one bound state then the same is true for h 0 v further if h 0 1 4 v 2 has infinitely many bound states then so does h 0 v consequences include the fact that for decaying potential v with lim inf n nv n 1 lim inf n nv n 1 h 0 v has infinitely many bound states the signs of v are irrelevant higher dimensional analogues are also discussed

invariants of 3 manifolds via link polynomials and quantum groups the aim of this paper is to construct new topological invariants of compact oriented 3 manifolds and of framed links in such manifolds our invariant of a link in a closed oriented 3 manifold is a sequence of complex numbers parametrized by complex roots of 1 for a framed link in s 3 the terms of the sequence are equale to the values of the suitably parametrized jones polynomial of the link in the corresponding roots of 1 in the case of manifolds with boundary our invariant is a sequence of finite dimensional complex linear operators this produces from each root of unity q a 3 dimensional topological quantum field theory

fusion categories arising from semisimple lie algebras using tilting modules we equip certain semisimple categories with a reduced tensor product structure the fusion rules for this tensor product are determined via known character formulas for the involved modules

existence and bifurcation of solutions for an elliptic degenerate problem abstract we investigate the existence multiplicity and bifurcation of solutions of a model nonlinear degenerate elliptic differential equation x 2 u u u p 1 u in 0 1 u 0 u 1 0 this model is related to a simplified version of the nonlinear wheeler dewitt equation as it appears in quantum cosmological models we prove the existence of multiple positive solutions more precisely we show that there exists an infinite number of connected branches of solutions which bifurcate from the bottom of the essential spectrum of the corresponding linear operator nous etudions ici l existence multiplicite et proprietes de bifurcation des solutions d un probleme elliptique degenere x 2 u u u p 1 u in 0 1 u 0 u 1 0 ce probleme modele est proche d une version simplifiee et non lineaire de l equation de wheeler dewitt utilisee dans des modeles de cosmologie quantique nous prouvons l existence d une infinite de branches de solutions qui bifurquent a partir de l infimum du spectre continu de l operateur lineaire correspondant

a lower bound for the number of negative eigenvalues of schrodinger operators we prove a lower bound for the number of negative eigenvalues for a schr o dinger operator on a riemannian manifold via the integral of the potential

eigenvalues of elliptic boundary value problems with an indefinite weight function on etudie les problemes aux valeurs propres elliptiques autoadjoints au r x u dans un ensemble ouvert r k a est un operateur positif d ordre 2m et le poids r est une fonction qui change de signe dans et peut etre discontinue on determine le comportement asymptotique des valeurs propres

lower bounds of eigenvalues for a class of bi subelliptic operators let be a bounded open domain in rnrn with smooth boundary and x x1 x2 xm x x1 x2 xm be a system of real smooth vector fields defined on with the boundary which is non characteristic for x if x satisfies the hormander s condition then the vector fields are finitely degenerate and the sum of square operators x i 1mxi2 is a subelliptic operator let k k be the k th eigenvalue for the bi subelliptic operator x2 on in this paper we introduce the generalized metivier s condition and study the lower bounds of dirichlet eigenvalues for the operator x2 on some finitely degenerate systems of vector fields x which satisfy the hormander s condition or the generalized metivier s condition by using the subelliptic estimates we shall give a explicit lower bound estimates of k k which is polynomial increasing in k with the order relating to the hormander index or the generalized metivier index

new bounds on the lieb thirring constants improved estimates on the constants l d for 1 2 3 2 d n in the inequalities for the eigenvalue moments of schrodinger operators are established

multidimensional lower bounds for the eigenvalues of stokes and dirichlet laplacian operators this article is concerned with the improvements of certain eigenvalue inequalities of stokes operator and dirichlet laplacian related to the berezin li yau type inequalities the formulas proved extend the earlier works of melas a lower bound for sums of eigenvalues of the laplacian proc am math soc 131 2 631 636 2002 10 1090 s0002 9939 02 06834 x on dirichlet laplacian and of ilyin lower bounds for the spectrum of the laplace and stokes operators discrete contin dyn syst 28 1 131 146 2010 10 3934 dcds 2010 28 131 on stokes operator for any dimension d 2 and they are asymptotically sharp as are the earlier inequalities of berezin li yau melas and ilyin

lieb thirring inequality with semiclassical constant and gradient error term abstract in 1975 lieb and thirring derived a semiclassical lower bound on the kinetic energy for fermions which agrees with the thomas fermi approximation up to a constant factor whenever the optimal constant in their bound coincides with the semiclassical one is a long standing open question we prove an improved bound with the semiclassical constant and a gradient error term which is of lower order

gold music a variation on music to accurately determine peaks of the spectrum this paper presents a new way of accurately determining peaks of the music multiple emitter location and signal parameter estimation r o schmidt ieee trans antennas and propagation vol ap 34 no 3 pp 276 280 mar 1986 spectrum here considered from the point of view of estimating the directions of arrival doas of narrowband signals it can be used with any smart antenna geometry and for any purpose where music is applicable the music algorithm for doa estimation evaluates the music spectrum for various angles and chooses the maxima or peaks as the angles of arrival the values obtained depend on the interval at which the spectrum is evaluated the coarser the interval the less accurate are the results in case of music to improve accuracy and not depend on the interval root music direction finding for diversely polarized signals using polynomial rooting a j weiss and b friedlander ieee trans signal processing vol 41 no 5 pp 1893 1905 may 1993 which involves finding the roots of a polynomial is available however root music is applicable in its original form only to uniform linear arrays ula the gold music algorithm proposed in this paper is a two stage process the first stage evaluates the objective function at coarse intervals and determines peaks followed by an iterative approach based on gold section univariate gsu minimization algorithms for minimization without derivatives r brent englewood cliffs nj usa prentice hall 1983 to find accurate values of the peaks if the number of peaks found so far is equal to the number of estimated peaks the algorithm stops with this first stage the second stage is an iterative step for fine resolution using finer intervals around the peaks found so far for finding peaks that were missing in previous iterations this paper also presents a method based on a partitioning algorithm for estimating the number of emitters the performance of gold music is described including its advantages and comparison of time complexities for music root music and gold music the proposed algorithm gives good results even when the number of snapshots is small this gives it an additional computational advantage it does not compromise on the resolving power of music

the eigenvalue problem and infinitely many sign changing solutions for an elliptic equation with critical hardy constant abstract by li yau s idea and a more precise hardy s inequality we obtain a new lower bound of eigenvalues for an elliptic equation with critical hardy constant then using the lower bounds we prove that there are infinitely many sign changing solutions for a class of elliptic equation with critical hardy constant which extends the results in schechter and zou 2005 9 to the critical hardy constant case

multiple emitter location and signal parameter estimation processing the signals received on an array of sensors for the location of the emitter is of great enough interest to have been treated under many special case assumptions the general problem considers sensors with arbitrary locations and arbitrary directional characteristics gain phase polarization in a noise interference environment of arbitrary covariance matrix this report is concerned first with the multiple emitter aspect of this problem and second with the generality of solution a description is given of the multiple signal classification music algorithm which provides asymptotically unbiased estimates of 1 number of incident wavefronts present 2 directions of arrival doa or emitter locations 3 strengths and cross correlations among the incident waveforms 4 noise interference strength examples and comparisons with methods based on maximum likelihood ml and maximum entropy me as well as conventional beamforming are included an example of its use as a multiple frequency estimator operating on time series is included

the thermodynamic limit of quantum coulomb systems part ii applications abstract in a previous paper c hainzl m lewin j p solovej the thermodynamic limit of quantum coulomb systems part i general theory adv math 2009 doi 10 1016 j aim 2008 12 010 this issue we have developed a general theory of thermodynamic limits we apply it here to three different coulomb quantum systems for which we prove the convergence of the free energy per unit volume the first system is the crystal for which the nuclei are classical particles arranged periodically in space and only the electrons are quantum particles we recover and generalize a previous result of fefferman in the second example both the nuclei and the electrons are quantum particles submitted to a periodic magnetic field we thereby extend a seminal result of lieb and lebowitz finally in our last example we take again classical nuclei but optimize their position to our knowledge such a system was never treated before the verification of the assumptions introduced in c hainzl m lewin j p solovej the thermodynamic limit of quantum coulomb systems part i general theory adv math 2009 doi 10 1016 j aim 2008 12 010 this issue uses several tools which have been introduced before in the study of large quantum systems in particular an electrostatic inequality of graf and schenker is one main ingredient of our new approach

schrodinger operators with few bound states we show that whole line schrodinger operators with finitely many bound states have no embedded singular spectrum in contradistinction we show that embedded singular spectrum is possible even when the bound states approach the essential spectrum exponentially fast we also prove the following result for one and two dimensional schrodinger operators h with bounded positive ground states given a potential v if both h v are bounded from below by the ground state energy of h then v 0

on the center of the small quantum group using the quantum fourier transform f we describe the block decomposition and multiplicative structure of a subalgebra z f z of the center of the small quantum group uqfin g at a root of unity it contains the known subalgebra z which is isomorphic to the algebra of characters of finite dimensional uqfin g modules we prove that the intersection z f z coincides with the annihilator of the radical of z applying representation theoretical methods we show that z surjects onto the algebra of endomorphisms of certain indecomposable projective modules over uqfin g in particular this leads to the conclusion that the center of uqfin g coincides with z f z in the case g sl2

convergence of finite element solutions of stochastic partial integro differential equations driven by white noise numerical approximation of a stochastic partial integro differential equation driven by a space time white noise is studied by truncating a series representation of the noise with finite element method for spatial discretization and convolution quadrature for time discretization sharp order convergence of the numerical solutions is proved up to a logarithmic factor numerical examples are provided to support the theoretical analysis

a note on the fractal dimension of attractors of dissipative dynamical systems inotherwords thehausdorandfractaldimensionshavethesameupperbound turningtotheapplicationswerstnotethatinalltheexamplesofpartialdierentialequationsconsideredin 8 thefunctionf d isconcave secondly inthetwoparticularexamples areaction diusionsystemandtwo dimensionalnavier stokessystem weobtainf d explicitlyand consequently weobtainexplicitupperboundsofthefractaldimensionoftheattractorintermsofthephysicalparameters finally we observe that theorem 2 1 can be obtained by a careful analysis ofthecorrespondingresultsin 8 however weprovetheorem2 1generalizingtotheinnite dimensionalcasethemethodof 2 wherethekaplan yorkeestimateforthefractaldimensionofthelorenzattractorhasbeenobtained 2 main estimatesupposethatasemigroups

on lieb thirring inequalities for higher order operators with critical and subcritical powers let i h l v denote the negative eigenvalues of the operatorh l u l u v 0 x d onl 2 d we prove the two sided estimate open image in new window we discuss bounds on the riesz means open image in new window

on some sharp spectral inequalities for schrodinger operators on semiaxis in this paper we obtain sharp lieb thirring inequalities for a schrodinger operator on semiaxis with a matrix potential and show how they can be used to other related problems among them are spectral inequalities on star graphs and spectral inequalities for schrodinger operators on half spaces with robin boundary conditions

on the center of quantized enveloping algebras abstract letube a quasitriangular hopf algebra one may use ther matrix ofuin order to construct scalar invariants of knots analogously reshetikhin wrote down tangle invariants which take their values in the center ofu reshetikhin s expressions thus define central elements inu we prove here an identity characterizing some of these elements whenuis a quantized enveloping algebra as an application we give a proof for a statement of faddeev reshetikhin and takhtadzhyan concerning the center of a quantized enveloping algebra

gevrey regularity for navier stokes equations under lions boundary conditions abstract the navier stokes system is considered in a compact riemannian manifold gevrey class regularity is proven under lions boundary conditions in 2d for the rectangle cylinder and hemisphere and in 3d for the rectangle the cases of the 2d sphere and 2d and 3d torus are also revisited

le spectre de certaines mosaiques poissoniennes du plan et l enveloppe convexe du pont brownien the spectral empirical function of a homogeneous isotropic poisson mosaic process is a functional of the perimeter of the convex hull of planar brownian bridge some geometrical identities follow

esprit estimation of signal parameters via rotational invariance techniques an approach to the general problem of signal parameter estimation is described the algorithm differs from its predecessor in that a total least squares rather than a standard least squares criterion is used although discussed in the context of direction of arrival estimation esprit can be applied to a wide variety of problems including accurate detection and estimation of sinusoids in noise it exploits an underlying rotational invariance among signal subspaces induced by an array of sensors with a translational invariance structure the technique when applicable manifests significant performance and computational advantages over previous algorithms such as mem capon s mlm and music

estimates for eigenvalues on riemannian manifolds abstract in this paper we investigate eigenvalues of the dirichlet eigenvalue problem of laplacian on a bounded domain in an n dimensional complete riemannian manifold m when m is an n dimensional euclidean space r n the conjecture of polya is well known the kth eigenvalue k of the dirichlet eigenvalue problem of laplacian satisfies k 4 2 n vol 2 n k 2 n for k 1 2 li and yau p li s t yau on the schrodinger equation and the eigenvalue problem comm math phys 88 1983 309 318 cf lieb e lieb the number of bound states of one body schrodinger operators and the weyl problem in proc sympos pure math vol 36 1980 pp 241 252 have given a partial solution for the conjecture of polya that is they have proved 1 k i 1 k i n n 2 4 2 n vol 2 n k 2 n for k 1 2 which is sharp in the sense of average in this paper we consider a general setting for complete riemannian manifolds we establish an analog of the li and yau s inequality for eigenvalues of the dirichlet eigenvalue problem of laplacian on a bounded domain in a complete riemannian manifold furthermore we obtain a universal inequality for eigenvalues of the dirichlet eigenvalue problem of laplacian on a bounded domain in a hyperbolic space h n 1 from it we prove that when the bounded domain tends to h n 1 all eigenvalues tend to n 1 2 4

liouville theorems for stable solutions of semilinear elliptic equations with convex nonlinearities abstract we extend our recent results on the classification of stable solutions of the equation u f u in r n where f 0 is a general convex non decreasing function

invariants of 3 manifolds and projective representations of mapping class groups via quantum groups at roots of unity an example of a finite dimensional factorizable ribbon hopf algebra is given by a quotienth u q g of the quantized universal enveloping algebrau q g at a root of unityq of odd degree the mapping class groupm g 1 of a surface of genusg with one hole projectively acts by automorphisms in theh moduleh g ifh is endowed with the coadjointh module structure there exists a projective representation of the mapping class groupm g n of a surface of genusg withn holes labeled by finite dimensionalh modulesx 1 x n in the vector space hom h x 1 x n h g an invariant of closed oriented 3 manifolds is constructed modifications of these constructions for a class of ribbon hopf algebras satisfying weaker conditions than factorizability including most ofu q g at roots of unityq of even degree are described

weyl s type estimates on the eigenvalues of critical schrodinger operators using improved hardy sobolev inequalities motivated by the work karachalios n i 2008 lett math phys 83 189 99 we present explicit asymptotic estimates on the eigenvalues of the critical schrodinger operator involving inverse square potential based on improved hardy sobolev type inequalities

analysis of the asymptotic relative efficiency of the music algorithm an analytical performance evaluation of the errors of the direction of arrival estimates obtained by the music algorithm for uncorrelated sources is provided explicit asymptotic formulas are derived for the means and the covariance of the estimates the covariances are then compared to the cramer rao lower bound it is shown that for a single course the music algorithm is asymptotically efficient for multiple sources the algorithm is not efficient in general however it approaches asymptotic efficiency when the snrs signal to noise ratios of all sources tend to infinity it is illustrated by several test cases that the relative efficiency of the music algorithm is nearly one under a wide range of parameter variations the analytic performance evaluation thus confirms empirical evidence to the excellent performance of the music algorithm for narrowband signals

direction finding for diversely polarized signals using polynomial rooting a direction finding algorithm for diversely polarized arrays that is based on polynomial rooting is presented using polynomial rooting instead of search reduces significantly the computational requirements of the algorithm and enhances resolution the basic method is limited to linear uniformly spaced arrays however using an interpolation technique the algorithm is extended to a larger class of arrays the performance of the proposed algorithm is analyzed computer simulations are used to demonstrate the performance of the new technique and to verify the performance analysis

multiplicity results for some perturbed elliptic problems in unbounded domains with non homogeneous boundary conditions abstract we study the following nonlinear elliptic problem u m u g x u f x in r n b r u h on b r u 0 as x with n 3 m r under suitable conditions on the radial functions g f and h multiplicity results are proved in the perturbed case f 0 and h 0 by using bolle s perturbation method and suitable growth estimates on min max critical levels

dirichlet and neumann eigenvalue problems on domains in euclidean spaces abstract we obtain here some inequalities for the eigenvalues of dirichlet and neumann value problems for general classes of operators or system of operators acting in l 2 or l 2 c m r d d 1

steklov eigenvalues and quasiconformal maps of simply connected planar domains we investigate isoperimetric upper bounds for sums of consecutive steklov eigenvalues of planar domains the normalization involves the perimeter and scale invariant geometric factors which measure deviation of the domain from roundness we prove sharp upper bounds for both starlike and simply connected domains for a large collection of spectral functionals including partial sums of the zeta function and heat trace the proofs rely on a special class of quasiconformal mappings

the number of bound states for a discrete schrodinger operator on bb z n n geq1 lattices we consider the discrete schrodinger operator in in the case of a potential with negative part in an appropriate l space decays with an appropriate rate we present a discrete analog of the method of li and yau 1983 commun math phys 88 309 18 proving an explicit upper estimate on the number of bound states which is independent of the dimension of the lattice this is a major difference with the continuous counterpart estimate which is not valid when n 1 2 as a consequence a dimension independent smallness criterion for the existence of bound states is derived in contrast to the continuous case as well as to the discrete case of vanishing potential a short comment is made on possible applications of the results to the study of the dynamics of some particular spatially discrete nonlinear systems

sharp two sided heat kernel estimates for critical schrodinger operators on bounded domains on a smooth bounded domain omega subset bf rm r n we consider the schrodinger operators v with v being either the critical borderline potential v x n 2 2 4 x 2 or v x 1 4 dist x 2 under dirichlet boundary conditions in this work we obtain sharp two sided estimates on the corresponding heat kernels to this end we transform the schrodinger operators into suitable degenerate operators for which we prove a new parabolic harnack inequality up to the boundary to derive the harnack inequality we have established a series of new inequalities such as improved hardy logarithmic hardy sobolev hardy moser and weighted poincare as a byproduct of our technique we are able to answer positively to a conjecture of e b davies

bounds on eigenvalues of dirichlet laplacian in this paper we investigate an eigenvalue problem of dirichlet laplacian on a bounded domain in an n dimensional euclidean space rn if k 1 is the k 1 th eigenvalue of dirichlet laplacian on then we prove that for n 41 and k geq 41 lambda k 1 leq k frac2n lambda 1 and for any n and k lambda k 1 leq c 0 n k k frac2n lambda 1 with c 0 n k leq j 2 n 2 1 j 2 n 2 1 1 where jp k denotes the k th positive zero of the standard bessel function jp x of the first kind of order p from the asymptotic formula of weyl and the partial solution of the conjecture of polya we know that our estimates are optimal in the sense of order of k

laplacians on periodic graphs with guides abstract we consider laplace operators on periodic discrete graphs perturbed by guides i e graphs which are periodic in some directions and finite in other ones the spectrum of the laplacian on the unperturbed graph is a union of a finite number of non degenerate bands and eigenvalues of infinite multiplicity we show that the spectrum of the perturbed laplacian consists of the unperturbed one plus the additional so called guided spectrum which is a union of a finite number of bands we estimate the position of the guided bands and their length in terms of geometric parameters of the graph we also determine the asymptotics of the guided bands for guides with large multiplicity of edges moreover we show that the possible number of guided bands their length and position can be rather arbitrary for some specific periodic graphs with guides

wesd weighted spectral distance for measuring shape dissimilarity this paper presents a new distance for measuring shape dissimilarity between objects recent publications introduced the use of eigenvalues of the laplace operator as compact shape descriptors here we revisit the eigenvalues to define a proper distance called weighted spectral distance wesd for quantifying shape dissimilarity the definition of wesd is derived through analyzing the heat trace this analysis provides the proposed distance with an intuitive meaning and mathematically links it to the intrinsic geometry of objects we analyze the resulting distance definition present and prove its important theoretical properties some of these properties include 1 wesd is defined over the entire sequence of eigenvalues yet it is guaranteed to converge 2 it is a pseudometric 3 it is accurately approximated with a finite number of eigenvalues and 4 it can be mapped to the 0 1 interval last experiments conducted on synthetic and real objects are presented these experiments highlight the practical benefits of wesd for applications in vision and medical image analysis

global bounds of schrodinger heat kernels with negative potentials abstract we obtain global in time bounds for the heat kernel g of the schrodinger operator l v the potential v satisfies v x c d x b near infinity with b 0 the result can be described as follows suppose l is positive and b 2 then g g x t y 0 has a global upper bound which is a standard gaussian times a polynomial function of x y t however when l is not nonnegative and negative eigenvalues exist g g x t 0 0 is bounded below by a standard gaussian times ectwithc 0 in other words the growth rate of g x t 0 0 is comparable with the heat kernel of c regardless how fast v decays near infinity

global existence and compact attractors for the discrete nonlinear schrodinger equation abstract we study the asymptotic behavior of solutions of discrete nonlinear schrodinger type dnls equations for a conservative system we consider the global in time solvability and the question of existence of standing wave solutions similarities and differences with the continuous counterpart nls partial differential equation are pointed out for a dissipative system we prove existence of a global attractor and its stability under finite dimensional approximations similar questions are treated in a weighted phase space finally we propose possible extensions for various types of dnls equations

sums of laplace eigenvalues rotations and tight frames in higher dimensions the sum of the first n 1 eigenvalues of the laplacian is shown to be maximal among simplexes for the regular simplex the regular tetrahedron in three dimensions maximal among parallelepipeds for the hypercube and maximal among ellipsoids for the ball provided the volume and moment of inertia of an auxiliary body are suitably normalized this result holds for dirichlet robin and neumann eigenvalues additionally the cubical torus is shown to be maximal among flat tori the proof involves euclidean tight frames generated by orbits of the rotation group of the extremal domain among general convex domains the ball is conjectured to maximize sums of neumann eigenvalues provided the volume and moment of inertia of the polar dual are suitably normalized

on the schrodinger equation and the eigenvalue problem if k is thekth eigenvalue for the dirichlet boundary problem on a bounded domain in n h weyl s asymptotic formula asserts that lambda k sim c n left frac k v d right 2 n hence sum limits i 1 k lambda i sim frac nc n n 2 k frac n 2 n v d 2 n we prove that for any domain and for all sum limits i 1 k lambda i geqq frac nc n n 2 k frac n 2 n v d 2 n a simple proof for the upper bound of the number of eigenvalues less than or equal to for the operator v x defined on n n 3 in terms of int limits mathbb r n v alpha n 2 is also provided

performance analysis of root music the authors analyze the performance of root music a variation of the music algorithm for estimating the direction of arrival doa of plane waves in white noise in the case of a linear equispaced sensor array the performance of the method is analyzed by examining the perturbation in the roots of the polynomial formed in the intermediate step of root music in particular asymptotic results for the mean squared error in the estimates of the direction of arrival are derived simplified expressions are presented for the one and two source case and compared to those obtained for least squares esprit computer simulations are also presented and they are in close agreement with the theory an important outcome of this analysis is the fact that the error in the signal zeros has a largely radial component this provides an explanation as to why the root music is superior to the spectral music algorithm

nonlocal operators of order near zero abstract we study dirichlet forms defined by nonintegrable levy kernels whose singularity at the origin can be weaker than that of any fractional laplacian we show some properties of the associated sobolev type spaces in a bounded domain such as symmetrization estimates hardy inequalities compact inclusion in l 2 or the inclusion in some lorentz space we then apply those properties to study the associated nonlocal operator l and the dirichlet and neumann problems related to the equations l u f x and l u f u in

qualitative and quantitative estimates for minimal hypersurfaces with bounded index and area we prove qualitative estimates on the total curvature of closed minimal hypersurfaces in closed riemannian manifolds in terms of their index and area restricting to the case where the hypersurface has dimension less than seven in particular we prove that if we are given a sequence of closed minimal hypersurfaces of bounded area and index the total curvature along the sequence is quantised in terms of the total curvature of some limit surface plus a sum of total curvatures of complete properly embedded minimal hypersurfaces in euclidean space all of which are finite thus we obtain qualitative control on the topology of minimal hypersurfaces in terms of index and area as a corollary

effective limiting absorption principles and applications the limiting absorption principle asserts that if h is a suitable schrodinger operator and f lives in a suitable weighted l2 space namely h 0 1 2 sigma for some sigma 0 then the functions r lambda i varepsilon f h lambda i varepsilon 1 f converge in another weighted l2 space h 0 1 2 sigma to the unique solution u of the helmholtz equation h lambda u f which obeys the sommerfeld outgoing radiation condition in this paper we investigate more quantitative or effective versions of this principle for the schrodinger operator on asymptotically conic manifolds with short range potentials and in particular consider estimates of the form r lambda i varepsilon f h 0 1 2 sigma leq c lambda h f h 0 1 2 sigma we are particularly interested in the exact nature of the dependence of the constants c lambda h on both lambda and h it turns out that the answer to this question is quite subtle with distinctions being made between low energies lambda ll 1 medium energies lambda sim 1 and large energies lambda gg 1 and there is also a non trivial distinction between qualitative estimates on a single operator h possibly obeying some spectral condition such as non resonance or a geometric condition such as non trapping and quantitative estimates which hold uniformly for all operators h in a certain class using elementary methods integration by parts and ode techniques we give some sharp answers to these questions as applications of these estimates we present a global in time local smoothing estimate and pointwise decay estimates for the associated time dependent schrodinger equation as well as an integrated local energy decay estimate and pointwise decay estimates for solutions of the corresponding wave equation under some additional assumptions on the operator h

bounds on the number of bound states for potentials with critical decay at infinity let v be a potential whose negative part v decays like c x 2 at infinity if c is not too large then the schrodinger operator hv v on rn n 3 has only a finite number of bound states although the associated classical phase space volume is infinite optimal conditions are derived for the absence of bound states and a family of bounds on the total number of bound states for the operator hv of the form n0 v c n rn v c x 2 x x0 2 n dn x the basic tool used to obtain results is a family of sharp sobolev inequalities for the operator a c c x 2

a unified approach to improved l p hardy inequalities with best constants we present a unified approach to improved l p hardy inequalities in r n we consider hardy potentials that involve either the distance from a point or the distance from the boundary or even the intermediate case where the distance is taken from a surface of codimension 1 k n in our main result we add to the right hand side of the classical hardy inequality a weighted l p norm with optimal weight and best constant we also prove nonhomogeneous improved hardy inequalities where the right hand side involves weighted l q norms q p

asymptotic behaviour of extremal averages of laplacian eigenvalues we study the convergence of extrema of averages of eigenvalues of the dirichlet laplacian on domains in mathbb r n under both measure and surface measure restrictions in the former case we prove that the sequence of averages to the power n 2 is sub additive and determine the first term in its asymptotics in the high frequency limit in the latter case we show that the sequence of minimisers converges to the ball as the frequency goes to infinity similar results hold for neumann boundary conditions

a bound for the eigenvalue counting function for krein von neumann and friedrichs extensions for an arbitrary open nonempty bounded set rn n n and sufficiently smooth coefficients a b q we consider the closed strictly positive higher order differential operator a 2m a b q in l2 defined on w02m 2 associated with the differential expression 2m a b q j k 1n i j bj aj k i k bk q m m n and its krein von neumann extension ak 2m a b q in l2 denoting by n ak 2m a b q 0 the eigenvalue counting function corresponding to the strictly positive eigenvalues of ak 2m a b q we derive the bound n ak 2m a b q cvn 2 n 1 2m2m n n 2m n 2m 0 where c c a b q 0 with c in 0 0 is connected to the eigenfunction expansion of the self adjoint operator a 2m a b q in l2 rn defined on w2m 2 rn corresponding to 2m a b q here vn n 2 n 2 2 denotes the euclidean volume of the unit ball in rn our method of proof relies on variational considerations exploiting the fundamental link between the krein von neumann extension and an underlying abstract buckling problem and on the distorted fourier transform defined in terms of the eigenfunction transform of a 2 a b q in l2 rn we also consider the analogous bound for the eigenvalue counting function for the friedrichs extension af 2m a b q in l2 of a 2m a b q no assumptions on the boundary of are made

lyapunov stability of ground states of nonlinear dispersive evolution equations on presente une nouvelle demonstration de la stabilite orbitale des solitons d etat fondamental de l equation de schrodinger non lineaire pour une grande classe de non linearites on demontre la stabilite de l onde solitaire pour l equation de korteweg de vries generalisee

music maximum likelihood and cramer rao bound the performance of the music and ml methods is studied and their statistical efficiency is analyzed the cramer rao bound crb for the estimation problems is derived and some useful properties of the crb covariance matrix are established the relationship between the music and ml estimators is investigated as well a numerical study is reported of the statistical efficiency of the music estimator for the problem of finding the directions of two plane waves using a uniform linear array an exact description of the results is included

tensor products of quantized tilting modules letuk denote the quantized enveloping algebra corresponding to a finite dimensional simple complex lie algebral assume that the quantum parameter is a root of unity ink of order at least the coxeter number forl also assume that this order is odd and not divisible by 3 if typeg2 occurs we demonstrate how one can define a reduced tensor product on the familyf consisting of those finite dimensional simpleuk modules which are deformations of simplel and which have non zero quantum dimension this together with the work of reshetikhin turaev and turaev wenzl prove that uk f is a modular hopf algebra and hence produces invariants of 3 manifolds also by recent work of duurhus jakobsen and nest it leads to a general topological quantum field theory the method of proof explores quantized analogues of tilting modules for algebraic groups

estimates on the eigenvalues of the clamped plate problem on domains in euclidean spaces the purpose of this article is two fold first we obtain some certain bounds for the sums of positive and negative powers of the eigenvalues of the clamped plate problem of the dirichlet bi laplacian operator 2 d restricted to a bounded domain d rd with d 2 second we establish lower bounds for the sums of eigenvalues of 2 d sharper than the bounds recently obtained by cheng and wei a lower bound for eigenvalues of a clamped plate problem calculus var partial differ equ 42 3 4 579 590 2011 10 1007 s00526 011 0399 6 all these estimates are sharp in the sense of weyl asymptotics

universal bounds for eigenvalues of a buckling problem in this paper we investigate an eigenvalue problem for a biharmonic operator on a bounded domain in an n dimensional euclidean space which is also called a buckling problem we introduce a new method to construct nice trial functions and we derive a universal inequality for higher eigenvalues of the buckling problem by making use of the trial functions thus we give an affirmative answer for the problem on universal bounds for eigenvalues of the buckling problem which was proposed by payne polya and weinberger in 14 and this problem has been mentioned again by ashbaugh in 1

sums of laplace eigenvalues rotationally symmetric maximizers in the plane abstract the sum of the first n 1 eigenvalues of the laplacian is shown to be maximal among triangles for the equilateral triangle maximal among parallelograms for the square and maximal among ellipses for the disk provided the ratio area 3 moment of inertia for the domain is fixed this result holds for both dirichlet and neumann eigenvalues and similar conclusions are derived for robin boundary conditions and schrodinger eigenvalues of potentials that grow at infinity a key ingredient in the method is the tight frame property of the roots of unity for general convex plane domains the disk is conjectured to maximize sums of neumann eigenvalues

index energy estimates for yang mills connections and einstein metrics we prove a conformally invariant estimate for the index of schrodinger operators acting on vector bundles over four manifolds related to the classical cwikel lieb rozenblum estimate applied to yang mills connections we obtain a bound for the index in terms of its energy which is conformally invariant and captures the sharp growth rate furthermore we derive an index estimate for einstein metrics in terms of the topology and the einstein hilbert energy lastly we derive conformally invariant estimates for the betti numbers of an oriented four manifold with positive scalar curvature

infinitely many sign changing solutions for a class of biharmonic equation without symmetry abstract in this paper via a perturbation theory we prove the existence of infinitely many sign changing solutions for a class of biharmonic equations

characters and a verlinde type formula for symmetric hopf algebras abstract we study certain aspects of finite dimensional non semisimple symmetric hopf algebras h and their duals h we focus on the set i h of characters of projective h modules which is an ideal of the algebra of cocommutative elements of h this ideal corresponds via a symmetrizing form to the projective center higman ideal of h which turns out to be h where is an integral of h and is the left adjoint action of h on itself we describe h via primitive and central primitive idempotents of h we also show that it is stable under the quantum fourier transform our best results are obtained when h is a factorizable ribbon hopf algebra over an algebraically closed field of characteristic 0 in this case h is also the image of i h under a translated drinfel d map we use this fact to prove the existence of a steinberg like character the above ingredients are used to prove a verlinde type formula for h

optimal spectral rectangles and lattice ellipses we consider the problem of minimizing the k th eigenvalue of rectangles with unit area and dirichlet boundary conditions this problem corresponds to finding the ellipse centred at the origin with axes on the horizontal and vertical axes with the smallest area containing k integer lattice points in the first quadrant we show that as k goes to infinity the optimal rectangle approaches the square and correspondingly the optimal ellipse approaches the circle we also provide a computational method for determining optimal rectangles for any k and relate the rate of convergence to the square with the conjectured error term for gauss s circle problem

braided groups and quantum fourier transform abstract we show that acting on every finite dimensional factorizable ribbon hopf algebra h there are invertible operators s t obeying the modular identities s t 3 s 2 where is a constant the class includes the finite dimensional quantum groups uq g associated to complex simple lie algebras we give the example of uq sl 2 at a root of unity in detail as well as an example relating to anyons the operator s plays the role of quantum fourier transform and acts naturally on h viewed by transmutation as a braided group h a braided cocommutative hopf algebra in a braided category it obeys s 2 s 1 where s is the antipode of h the results follow as an application of previous category theoretical constructions