quantum unique ergodicity for sl2 z h we eliminate the possibility of escape of mass for hecke maass forms of large eigenvalue for the modular group combined with the work of lindenstrauss this establishes the quantum unique ergodicity conjecture of rudnick and sarnak for hecke maass forms on the modular surface sl 2

quantum variance and ergodicity for the baker s map we prove an egorov theorem or quantum classical correspondence for the quantised baker s map valid up to the ehrenfest time this yields a logarithmic upper bound for the decay of the quantum variance and as a corollary a quantum ergodic theorem for this map

kubik an automatic three dimensional finite element mesh generator this paper describes an automatic mesh generator for finite element problems in three dimensions called kubik kubik is based on a modular concept first a set of modules is generated each with its own mesh inside then the modules are placed in contact and a matrix describing the connectivity of the entire mesh is determined finally the geometry of the mesh is specified and all nodal co ordinates are determined by the computer the user s labour required to specify a mesh which represents a certain physical configuration resumbles the labour involved when the configuration has to be specified to the machine shop workers who are supposed to build it the modules are independent except for the compatibility requirements between modules which are to be placed in contact structures of any degree of complexity can in principle be obtained using kubik only two types of modules are required except when helicoidal structures are to be generated in which case it is convenient to use a third type the finite elements used are distorted cubes but they can be divided into tetrahedra prisms of pyramids nodes are placed at the corners of the cubes and additional nodes can be introduced if desired in order to obtain other types of finite elements in use infinite elements of quadrilateral cross section are also a standard feature of kubik and can be divided in order to obtain infinite prisms of triangular cross section kubik can also generate two dimensional meshes including meshing of a three dimensional surface for use with boundary elements

entropic bounds on semiclassical measures for quantized one dimensional maps quantum ergodicity asserts that almost all infinite sequences of eigenstates of quantized ergodic hamiltonian systems are equidistributed in phase space this however does not prohibit existence of exceptional sequences which might converge to different non liouville classical invariant measures it has been recently shown by n anantharaman and s nonnenmacher in 20 21 with h koch that for anosov geodesic flows the metric entropy of any semiclassical measure must satisfy a certain bound this remarkable result seems to be optimal for manifolds of constant negative curvature but not in the general case where it might become even trivial if the negative curvature of the riemannian manifold varies a lot it has been conjectured by the same authors that in fact a stronger bound valid in the general case should hold

quantum ergodicity and symmetry reduction abstract we study the spectral and ergodic properties of schrodinger operators on a compact connected riemannian manifold m without boundary in case that the examined system possesses certain symmetries more precisely if m carries an isometric and effective action of a compact connected lie group g we prove a generalized equivariant version of the semiclassical weyl law with an estimate for the remainder using a semiclassical functional calculus for h dependent functions and relying on recent results on singular equivariant asymptotics we then deduce an equivariant quantum ergodicity theorem under the assumption that the symmetry reduced hamiltonian flow on the principal stratum of the singular symplectic reduction of m is ergodic in particular we obtain an equivariant version of the shnirelman zelditch colin de verdiere theorem as well as a representation theoretic equidistribution theorem if m g is an orbifold similar results were recently obtained by kordyukov when g is trivial one recovers the classical results

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

hybrid subconvexity bounds for l left frac 1 2 hbox sym 2 f otimes g right fix an integer kappa geqslant 2 let p be prime and let k kappa be an even integer for f a holomorphic cusp form of weight k and full level and g a primitive holomorphic cusp form of weight 2 kappa and level p we prove hybrid subconvexity bounds for l left frac 1 2 text sym 2 f otimes g right in the k and p aspects when p frac 13 64 delta k p frac 3 8 delta for any 0 delta frac 11 128 these bounds are achieved through a first moment method with amplification when p frac 13 64 k leqslant p frac 4 13

scarred eigenstates for quantum cat maps of minimal periods in this paper we construct a sequence of eigenfunctions of the quantum arnold s cat map that in the semiclassical limit shows a strong scarring phenomenon on the periodic orbits of the dynamics more precisely those states have a semiclassical limit measure that is the sum of 1 2 the normalized lebesgue measure on the torus plus 1 2 the normalized dirac measure concentrated on any a priori given periodic orbit of the dynamics it is known the schnirelman theorem that most sequences of eigenfunctions equidistribute on the torus the sequences we construct therefore provide an example of an exception to this general rule our method of construction and proof exploits the existence of special values of h for which the quantum period of the map is relatively short and a sharp control on the evolution of coherent states up to this time scale we also provide a pointwise description of these states in phase space which uncovers their hyperbolic structure in the vicinity of the fixed points and yields more precise localization estimates

l infty norms of eigenfunctions of arithmetic surfaces here and elsewhere a l1o0 as a xc is a basic problem of quantum chaos s in any case almost nothing beyond 0 2 is known about 110 joc when the curvature is negative one can push the standard techniques and replace a1 4 by a1 4 log a in this case in this paper we use arithmetic techniques in particular modular correspondences to obtain the first improvement in the exponent 0 2 we also obtain lower bounds on the l norms in these arithmetic cases in more detail let a b be a quaternion division algebra over q a is linearly generated by 1 w q wq over q and w2 a q2 b wq qw 0 here a b e z are square free and we will assume that a 0 as usual the norm and trace are defined by n a a d tr a a oy where if a xo x1w x2q x3wq

the quantum unique ergodicity conjecture for thin sets abstract we consider some analogs of the quantum unique ergodicity conjecture for geodesics and shrinking families of sets in particular we prove the analog of the que conjecture for eisenstein series restricted to the infinite geodesic connecting 0 and inside the modular surface

estimates for rankin selberg l functions and quantum unique ergodicity abstract subconvex bounds in the weight aspect for rankin selberg l functions associated to two cusp forms are established these bounds are applied to prove the equidistribution of mass conjecture for cm forms

entropy of semiclassical measures in dimension 2 we study the high energy asymptotic properties of eigenfunctions of the laplacian in the case of a compact riemannian surfacem of anosov type to do this we look at families of distributions associated to them on the cotangent bundle t m and we derive entropic properties on their accumulation points in the high energy limit the so called semiclassical measures we show that the kolmogorov sinai entropy of a semiclassical measure for the geodesic ow gt is bounded from below by half of the ruelle upper bound i e hks g 1 2 s m d where is the upper lyapunov exponent at point

fractal uncertainty principle with explicit exponent we prove an explicit formula for the dependence of the exponent beta in the fractal uncertainty principle of bourgain dyatlov ann math 187 1 43 2018 on the dimension delta and on the regularity constant c r for the regular set in particular this implies an explicit essential spectral gap for convex co compact hyperbolic surfaces when the hausdorff dimension of the limit set is close to 1

a fast and robust cluster update algorithm for image segmentation in spin lattice models without annealing visual latencies revisited image segmentation in spin lattice models relies on the fast and reliable assignment of correct labels to those groups of spins that represent the same object commonly used local spin update algorithms are slow because in each iteration only a single spin is flipped and a careful annealing schedule has to be designed in order to avoid local minima and correctly label larger areas updating of complete spin clusters is more efficient but often clusters that should represent different objects will be conjoined in this study we propose a cluster update algorithm that similar to most local update algorithms calculates an energy function and determines the probability for flipping a whole cluster of spins by the energy gain calculated for a neighborhood of the regarded cluster the novel algorithm called energy based cluster update ecu algorithm is compared to its predecessors a convergence proof is derived and it is shown that the algorithm outperforms local update algorithms by far in speed and re

equidistribution of cusp forms in the level aspect let f traverse a sequence of classical holomorphic newforms of fixed weight and increasing square free level q infinity we prove that the pushforward of the mass off to the modular curve of level 1 equidistributes with respect to the poincare measure

equidistribution of eisenstein series in the level aspect we prove an equidistribution property of the eisenstein series for congruence subgroups as the level goes to infinity this is an analogy of the phenomenon called quantum ergodicity

arithmetic and equidistribution of measures on the sphere motivated by problems of mathematical physics quantum chaos questions of equidistribution of eigenfunctions of the laplace operator on a riemannian manifold have been studied by several authors we consider here in analogy with arithmetic hyperbolic surfaces orthonormal bases of eigenfunctions of the laplace operator on the two dimensional unit sphere which are also eigenfunctions of an algebra of hecke operators which act on these spherical harmonics we formulate an analogue of the equidistribution of mass conjecture for these eigenfunctions as well as of the conjecture that their moments tend to moments of the gaussian as the eigenvalue increases for such orthonormal bases we show that these conjectures are related to the analytic properties of degree eight arithmetic l functions associated to triples of eigenfunctions moreover we establish the conjecture for the third moments and give a conditional on standard analytic conjectures about these arithmetic l functions proof of the equidistribution of mass conjecture

quantum unique ergodicity of degenerate eisenstein series on gl n in the area of quantum chaos it is of great interest to study the distribution of the l 2 mass of eigenfunctions of the laplacian as eigenvalues tend to infinity luo and sarnak first formulated and proved arithmetic quantum unique ergodicity for the continuous spectrum spanned by eisenstein series of the hyperbolic laplacian on sl 2 mathbb z backslash mathbb h by utilizing the sub convexity bounds of l functions associated to maass cusp forms in this paper we build on luo and sarnak s method and explore the structure properties of the constant terms of gl n eisenstein series and extend their results to higher ranks we prove quantum unique ergodicity for a subspace of the continuous spectrum spanned by the degenerate eisenstein series on gl n

equidistribution of zeros of holomorphic sections in the non compact setting we consider tensor powers l n of a positive hermitian line bundle l h l over a non compact complex manifold x in the compact case b shiffman and s zelditch proved that the zeros of random sections become asymptotically uniformly distributed as n with respect to the natural measure coming from the curvature of l under certain boundedness assumptions on the curvature of the canonical line bundle of x and on the chern form of l we prove a non compact version of this result we give various applications including the limiting distribution of zeros of cusp forms with respect to the principal congruence subgroups of sl 2 and to the hyperbolic measure the higher dimensional case of arithmetic quotients and the case of orthogonal polynomials with weights at infinity we also give estimates for the speed of convergence of the currents of integration on the zero divisors

hecke theory and equidistribution for the quantization of linear maps of the torus 1 1 background one of the key issues of quantum chaos is the nature of the semiclassical limit of eigenstates of classically chaotic systems when the classical system is given by the geodesic flow on a compact riemannian manifoldm or rather on its cotangent bundle one can formulate the problem as follows the quantum hamiltonian is in suitable units represented by the positive laplacian on m to measure the distribution of its eigenstates we start with a smooth classical observable that is a smooth function on the unit cotangent bundle s m via some choice of quantization from symbols to pseudodifferential operators we form its quantization op f this is a zero order pseudodifferential operator with principal symbol f the expectation value of op f in the eigenstate is op f let j be a sequence of normalized eigenfunctions j j j 0 m j 2 1 the problem then is to understand the possible limits as j of the distributions f c s m op f j j 1 1

semiclassical limits for the hyperbolic plane we study the concentration properties of high energy eigenfunctions for the laplace beltrami operator of the hyperbolic plane with special consideration of automorphic eigenfunctions at the center of our investigation is the microlocal lift of an eigenfunction to sl 2 introduced by s zelditch the microlocal lift is based on s helgason s fourier transform and has a straightforward description in terms of the lie algebra sl 2 we begin with an elementary demonstration of zelditch s exact differential equation for the microlocal lift further for a sequence of suitably bounded eigenfunctions with eigenvalues tending to infinity we show that the limit of microlocal lifts is a geodesic flow invariant positive measure our main consideration is the microlocal lifts of the elementary eigenfunctions constructed from the macdonald bessel functions we find that with a scaling of auxiliary parameters the corresponding high energy limit converges to the positive dirac measure for the lift to sl 2 of a single geodesic on the upper half plane in particular at high energy the microlocal lift of a macdonald bessel function is concentrated along the lift of a single geodesic we further find that the convergence is uniform in parameters and thus our considerations can be applied to study automorphic eigenfunctions in particular the maass cusp forms for certain cofinite subgroups of sl 2 a formula is presented for the microlocal lift of a cusp form as an integral of a family of dirac measures and a measure given directly by sums of products of the classical fourier coefficients of the cusp form the formula establishes that questions on quantum chaos for automorphic eigenfunctions are equivalent to classically stated questions on twisted sums of coefficients in particular we find that the uniform distribution for the microlocal limit is equivalent to a uniform distribution for the limit of twisted coefficient sums we extend the considerations to the case of the nonholomorphic eisenstein series for the modular group we find that a result of w luo and p sarnak is equivalent to a limit sum formula involving the riemann zeta function and the elementary divisor function the formula is suggestive of a formula of s ramanujan

mapped infinite elements for exterior wave problems recently a new type of infinite elements which uses r 1 decay was proposed they were applied to exterior wave problems and good results were obtained in two dimensional problems however it was necessary to move the origin of the r 1 decay in order to model the outgoing wave more accurately because it decays roughly as r 1 2 in this paper the mapped infinite elements with r 1 2 decay and the necessary numerical integration procedure are presented these elements do not require any artificial movement of the origin several example problems are solved the results show that the infinite elements with r 1 2 decay here give much more accurate values than the infinite elements with exponential decay and any damper elements

local semicircle law for random regular graphs we consider random d regular graphs on n vertices with degree d at least log n 4 we prove that the green s function of the adjacency matrix and the stieltjes transform of its empirical spectral measure are well approximated by wigner s semicircle law down to the optimal scale given by the typical eigenvalue spacing up to a logarithmic correction aside from well known consequences for the local eigenvalue distribution this result implies the complete isotropic delocalization of all eigenvectors and a probabilistic version of quantum unique ergodicity 2017 wiley periodicals inc

random weighted sobolev inequalities and application to quantum ergodicity this paper is a continuation of poiret et al ann henri poincare 16 651 689 2015 where we studied a randomisation method based on the laplacian with harmonic potential here we extend our previous results to the case of any polynomial and confining potential v on mathbb r d we construct measures under concentration type assumptions on the support of which we prove optimal weighted sobolev estimates on mathbb r d this construction relies on accurate estimates on the spectral function in a non compact configuration space then we prove random quantum ergodicity results without specific assumption on the classical dynamics finally we prove that almost all bases of hermite functions are quantum uniquely ergodic

number variance for arithmetic hyperbolic surfaces we prove that the number variance for the spectrum of an arithmetic surface is highly nonrigid in part of the universal range in fact it is close to having a poisson behavior this fact was discovered numerically by schmit bogomolny georgeot and giannoni it has its origin in the high degeneracy of the length spectrum first observed by selberg

quantisations of piecewise parabolic maps on the torus and their quantum limits for general quantum systems the semiclassical behaviour of eigenfunctions in relation to the ergodic properties of the underlying classical system is quite difficult to understand the wignerfunctions of eigenstates converge weakly to invariant measures of the classical system the so called quantum limits and one would like to understand which invariant measures can occur that way thereby classifying the semiclassical behaviour of eigenfunctions

an infinite element and a formula for numerical quadrature over an infinite interval in this publication we present a new infinite element and discuss a formula for numerical quadrature over infinite regions both the element and the formula are based on the properties of a set of orthonormal functions obtained by mapping the legendre polynomials on the infinite interval the element can represent any physical function over the entire infinite region and the formula can integrate most integrands over the infinite interval they are thus very convenient for implementation in a general computer program which could then be used to solve different problems with no restrictions concerning the asymptotic behaviour of the solutions or the form of integrands to be integrated over the infinite interval however the element and the formula will perform at their best when used in conjunction and for problems of electrostatic magnetostatic etc potential the formula was compared with other existing formulae and found to be either superior or satisfactory in practically all cases as shown by the results of many numerical experiments expressions of shape functions are given for the element a test problem was solved using the element and the formula and the results are shown to be more accurate than those obtained when solving the same problem using infinite elements with exponential decay and gauss laguerre numerical quadrature

the modulus of continuity for γ0 m semi classical limits we study the behavior of a large eigenvalue limit of eigenfunctions for the hyperbolic laplacian for the modular quotient sl 2 fejer summation and results of s zelditch are used to show that the microlocal lifts of eigenfunctions have large eigenvalue limit a geodesic flow invariant measure for the modular unit cotangent bundle the limit is studied for hecke maass forms joint eigenfunctions of the hecke operators and the hyperbolic laplacian the first modulus of continuity result is presented for the limit the singular concentration set of the limit cannot be a compact union of closed geodesics and measured geodesic laminations

weak subconvexity for central values of l functions we describe a general method to obtain weak subconvexity bounds for many classes of l functions we give several examples of our bound and our work has applications to a conjecture of rudnick and sarnak for the mass equidistribution of hecke eigenforms

application of infinite elements to phase change situations on deforming meshes in recent years progress has been made in applying moving and deforming mesh systems to phase change problems this allows the numerical attention where it is needed near the migrating phase change zone in spatially unbounded problems one hopes that numerically finite outer boundaries either escape significant activity or are automatically pushed further away as activity nears not infrequently this approach fails temperature activity often spreads more rapidly than phase change thereby reaching far boundaries stretching of the mesh by movement of far boundaries can challenge mesh control and cause ill conditioning in this paper the advantages of time dependent mesh adaption are enhanced by the joining of a new formulation for infinite elements to far boundaries this is accomplished through a co ordinate transformation within the framework of conventional 2 d quadratic biquadratic and linear quadratic elements standard 2 by 2 gauss legendre quadrature suffices throughout and normal galerkin finite element features are undisturbed including strict conservation of energy the formulation is independent of global co ordinates entails no restrictions on the unknown function and should be applicable to other problem types all test cases on quadrilateral and triangular grids show very significant improvements with infinite elements relative to comparable solution systems using strictly finite grids

energy level statistics of the hadamard gutzwiller ensemble abstract the statistical properties of the quantal energy levels of the hadamard gutzwiller ensemble whose classical members belong to the class of systems with hard chaos are investigated based on a sample of 4500 energy levels it is shown that the short range statistics as nearest neighbour spacing distributions are governed by the goe predictions of random matrix theory which was first surmised by wigner and by landau and smorodinsky for nuclear level statistics this result strengthens the hypothesis that quantum systems with chaotic classical counterpart display level repulsion as predicted by random matrix theory however the level statistics describing correlations over greater level distances deviate from the goe predictions which is explained as a simple consequrnce of the fact that the spectral rigidity 3 l introduced by dyson and mehta saturates non universally at a finite value for l in complete agreement with the semiclassical theory developed by berry

uniform distribution of eigenfunctions on compact hyperbolic surfaces on demontre que les fonctions propres k du laplacien sur une surface hyperbolique compacte x devient uniformement distribuee sur x quand k

arithmetic quantum unique ergodicity for symplectic linear maps of the multidimensional torus we look at the expectation values for quantized linear symplectic maps on the multidimensional torus and their distribu tion in the semiclassical limit we construct super scars that are stable under the arithmetic symmetries of the system and localize on invariant manifolds we show that these super scars exist only when there are isotropic rational subspaces invariant under the linear map in the case where there are no such scars we com pute the variance of the fluctuations of the matrix elements for the desymmetrized system and present a conjecture for their limiting distributions

on the approximate normality of eigenfunctions of the laplacian the main result of this paper is a bound on the distance between the distribution of an eigenfunction of the laplacian on a compact riemannian manifold and the gaussian distribution if x is a random point on a manifold m and f is an eigenfunction of the laplacian with l 2 norm one and eigenvalue then d tv f x z 2 e f x 2 f x 2 this result is applied to construct specific examples of spherical harmonics of arbitrary odd degree which are close to gaussian in distribution a second application is given to random linear combinations of eigenfunctions on flat tori

the behaviour of eigenstates of arithmetic hyperbolic manifolds in this paper we study some problems arising from the theory of quantum chaos in the context of arithmetic hyperbolic manifolds we show that there is no strong localization scarring onto totally geodesic submanifolds arithmetic examples are given which show that the random wave model for eigenstates does not apply universally in 3 degrees of freedom