stability and isolation phenomena for yang mills fields in this article a series of results concerning yang mills fields over the euclidean sphere and other locally homogeneous spaces are proved using differential geometric methods one of our main results is to prove that any weakly stable yang mills field overs4 with groupg su2 su3 oru2 is either self dual or anti self dual the analogous statement for so4 bundles is also proved the other main series of results concerns gap phenomena for yang mills fields as a consequence of the non linearity of the yang mills equations we can give explicitc0 neighbourhoods of the minimal yang mills fields which contain no other yang mills fields in this part of the study the nature of the groupg does not matter neither is the dimension of the base manifold constrained to be four

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

view variation of point set and line segment features the variation with respect to view of 2d features defined for projections of 3d point sets and line segments is studied it is established that general case view invariants do not exist for any number of points given true perspective weak perspective or orthographic projection models feature variation under the weak perspective approximation is then addressed though there are no general case weak perspective invariants there are special case invariants of practical importance those cited in the literature are derived from linear dependence relations and the invariance of this type of relation to linear transformation the variation with respect to view is studied for an important set of 2d line segment features the relative orientation size and position of one line segment with respect to another the analysis includes an evaluation criterion for feature utility in terms of view variation this relationship is a function of both the feature and the particular configuration of 3d line segments the use of this information in objection recognition is demonstrated for difficult discrimination tasks

entropy of quantum limits in this paper we show that any measure arising as a weak limit of microlocal lifts of eigenfunctions of the laplacian on certain arithmetic manifolds have dimension at least 11 9 and in particular all ergodic components of this measure with respect to the geodesic flow have positive entropy

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

quantum variance of maass hecke cusp forms in this paper we study quantum variance for the modular surface x gamma backslash mathbb h where gamma sl 2 mathbb z is the full modular group we evaluate asymptotically the quantum variance which is introduced by s zelditch and describes the fluctuations of a quantum observable it is shown that the quantum variance is equal to the classical variance of the geodesic flow on s x the unit cotangent bundle of x but twisted by the central value of the maass hecke l functions

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 of c dynamical systems we define a notion of quantum or non commutative ergodicity for a class ofc dynamical systems a g which we callquantized gns systems such a system possesses a natural classical limit state which induces a classical limit system by the gns construction the criterion for quantum ergodicity is that the time average a of an observable a a equals the space average a i plus an errork which is negligible in the classical limit we prove that ergodicity of is a sufficient condition for quantum ergodicity of a g if the classical limit system is abelian give a conditional converse and discuss a number of applications

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

recent progress on the quantum unique ergodicity conjecture we report on some recent striking advances on the quantum unique ergodicity or que conjecture concerning the distribution of large frequency eigenfunctions of the laplacian on a negatively curved manifold the account falls naturally into two categories the first concerns the general conjecture where the tools are more or less limited to micro local analysis and the dynamics of the geodesic flow the second is concerned with arithmetic such manifolds where tools from number theory and ergodic theory of flows on homogeneous spaces can be combined with the general methods to resolve the basic conjecture as well as its holomorphic analogue the recent account by zelditch zel4 covers some of the same material and his and our discussion complement each other nicely as he goes into more detail with the first category and we with the second this note is not meant to be a survey of these topics and the discussion is not chronological our aim is to expose these recent developments after introducing the necessary backround which places them in their proper context recent progress on que 2 1 the general que conjecture in figure 1 the domains e s and b are an ellipse a stadium and a barnett billiard table respectively superimposed on these are the densities of a consecutive sequence of high frequency eigenfunctions states modes of the laplacian that is they are solutions to 4 j j j 0 in 0 dirichlet boundary conditions j dxdy 1 0 here 4 divgrad 2 x2 2 y2 1 2 3 are the eigenvalues and the eigenfunctions are normalized to have unit l norm the sequences are of 12 consecutive modes around the 5600 eigenvalue they are ordered from left to right and then down and the grayscale represents the probability density with zero white and larger values darker

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

large supremum norms and small shannon entropy for hecke eigenfunctions of quantized cat maps this paper concerns the behavior of eigenfunctions of quantized cat maps and in particular their supremum norm we observe that for composite integer values of n the inverse of planck s constant some of the desymmetrized eigenfunctions have very small support and hence very large supremum norm we also prove an entropy estimate and show that our functions satisfy equality in this estimate in the case when n is a prime power with even exponent we calculate the supremum norm for a large proportion of all desymmetrized eigenfunctions and we find that for a given n there is essentially at most four different values these assume

concentration of symmetric eigenfunctions abstract in this article we examine the concentration and oscillation effects developed by high frequency eigenfunctions of the laplace operator in a compact riemannian manifold more precisely we are interested in the structure of the possible invariant semiclassical measures obtained as limits of wigner measures corresponding to eigenfunctions these measures describe simultaneously the concentration and oscillation effects developed by a sequence of eigenfunctions we present some results showing how to obtain invariant semiclassical measures from eigenfunctions with prescribed symmetries as an application of these results we give a simple proof of the fact that in a manifold of constant positive sectional curvature every measure which is invariant by the geodesic flow is an invariant semiclassical measure

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

joint quasimodes positive entropy and quantum unique ergodicity we study joint quasimodes of the laplacian and one hecke operator on compact congruence surfaces and give conditions on the orders of the quasimodes that guarantee positive entropy on almost every ergodic component of the corresponding semiclassical measures together with the measure classification result of lindenstrauss ann math 2 163 1 165 219 2006 this implies quantum unique ergodicity for such functions our result is optimal with respect to the dimension of the space from which the quasi mode is constructed we also study equidistribution for sequences of joint quasimodes of the two partial laplacians on compact irreducible quotients of mathbb h times mathbb h

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

projective reconstruction and invariants from multiple images this correspondence investigates projective reconstruction of geometric configurations seen in two or more perspective views and the computation of projective invariants of these configurations from their images a basic tool in this investigation is the fundamental matrix that describes the epipolar correspondence between image pairs it is proven that once the epipolar geometry is known the configurations of many geometric structures for instance sets of points or lines are determined up to a collineation of projective 3 space spl pscrsup 3 by their projection in two independent images this theorem is the key to a method for the computation of invariants of the geometry invariants of six points in spl pscrsup 3 and of four lines in spl pscrsup 3 are defined and discussed an example with real images shows that they are effective in distinguishing different geometrical configurations since the fundamental matrix is a basic tool in the computation of these invariants new methods of computing the fundamental matrix from seven point correspondences in two images or six point correspondences in three images are given

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

invariant measures and arithmetic quantum unique ergodicity we classify measures on the locally homogeneous space i sl 2 r l which are invariant and have positive entropy under the diagonal subgroup of sl 2 r and recurrent under l this classification can be used to show arithmetic quantum unique ergodicity for compact arithmetic surfaces and a similar but slightly weaker result for the finite volume case other applications are also presented in the appendix joint with d rudolph we present a maximal ergodic theorem related to a theorem of hurewicz which is used in theproof of the main result

bounds for eigenforms on arithmetic hyperbolic 3 manifolds on a family of arithmetic hyperbolic 3 manifolds of square free level we prove an upper bound for the sup norm of hecke maas cusp forms with a power saving over the local geometric bound simultaneously in the laplacian eigenvalue and the volume by a novel combination of diophantine and geometric arguments in a noncommutative setting we obtain bounds as strong as the best corresponding results on arithmetic surfaces

bounds for rankin selberg integrals and quantum unique ergodicity for powerful levels reference epfl article 202508doi 10 1090 s0894 0347 2013 00779 1view record in web of science record created on 2014 10 23 modified on 2017 12 10

ergodic billiards that are not quantum unique ergodic partially rectangular domains are compact two dimensional riemannian manifolds x either closed or with boundary that contain a flat rectangle or cylinder in this paper we are interested in partially rectangular domains with ergodic billiard flow examples are the bunimovich stadium the sinai billiard or donnelly surfaces we consider a one parameter family x t of such domains parametrized by the aspect ratio t of their rectangular part there is convincing theoretical and numerical evidence that the laplacian on x t with dirichlet or neumann boundary conditions is not quantum unique ergodic que we prove that this is true for all t in 1 2 excluding possibly a set of lebesgue measure zero this yields the first examples of ergodic billiard systems proven to be non que

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

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

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

maximum norms of chaotic quantum eigenstates and random waves abstract the growth of the maximum norms of quantum eigenstates of classically chaotic systems with increasing energy is investigated the maximum norms provide a measure for localization effects in eigenfunctions an upper bound for the maxima of random superpositions of random functions is derived for the random wave model this gives the bound c ln e in the semiclassical limit e the growth of the maximum norms of random waves is compared with the growth of the maximum norms of the eigenstates of six quantum billiards which are classically chaotic the maximum norms of these systems are numerically shown to be conform with the random wave model furthermore the distribution of the locations of the maximum norms is discussed

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

on the structure of yang mills fields in compactified minkowski space in this work the yang mills functional with gauge group su 2 in compactified minkowski space is reduced to a mechanical lagrangian which enables one to find new classes of exact dual and nonself dual solutions the technique used in the reduction allows a simple interpretation of meron type solutions the formalism is then used to produce a good simple analytic approximation to static solutions of skyrme s field equation relevant to low energy hadronic interactions

quantum unique ergodicity for half integral weight automorphic forms we investigate the analogue of the quantum unique ergodicity que conjecture for half integral weight automorphic forms assuming the generalized riemann hypothesis grh we establish que for both half integral weight holomorphic hecke cusp forms for 0 4 lying in kohnen s plus subspace and for half integral weight hecke maa cusp forms for 0 4 lying in kohnen s plus subspace by combining the former result along with an argument of rudnick it follows that under grh the zeros of these holomorphic hecke cusp equidistribute with respect to hyperbolic measure on 0 4 h as the weight tends to infinity

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

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

rankin selberg l functions in the level aspect keywords moments rankin selberg convolution level aspect convexity breaking reference tan article 2002 003doi 10 1215 s0012 7094 02 11416 1 record created on 2008 11 14 modified on 2017 05 12

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

the existence of nonminimal solutions to the yang mills equation with group rm su 2 on s 2 times s 2 and s 1 times s 3

trace formulae for three dimensional hyperbolic lattices and application to a strongly chaotic tetrahedral billiard abstract this paper is devoted to the quantum chaology of three dimensional systems a trace formula is derived for compact polyhedral billiards which tessellate the three dimensional hyperbolic space of constant negative curvature the exact trace formula is compared with gutzwiller s semiclassical periodic orbit theory in three dimensions and applied to a tetrahedral billiard being strongly chaotic geometric properties as well as the conjugacy classes of the defining group are discussed the length spectrum and the quantal level spectrum are numerically computed allowing the evaluation of the trace formula as is demonstrated in the case of the spectral staircase n e which in turn is successfully applied in a quantization condition

entropy and the localization of eigenfunctions we study the large eigenvalue limit for the eigenfunctions of the laplacian on a compact manifold of negative curvature in fact we only assume that the geodesic flow has the anosov property in the semi classical limit we prove that the wigner measures associated to eigenfunctions have positive metric entropy in particular they cannot concentrate entirely on closed geodesies 1 introduction statement of results we consider a compact riemannian manifold m of dimension d 2 and assume that the geodesic flow ter acting on the unit tangent bundle of m has a chaotic behaviour this refers to the asymptotic properties of the flow when time t tends to infinity ergodicity mixing hyperbolicity we assume here that the geodesic flow has the anosov property the main example being the case of negatively curved manifolds the words quantum chaos express the intuitive idea that the chaotic features of the geodesic flow should imply certain special features for the corresponding quantum dynamical system that is according to schrodinger the unitary flow exp iht t r acting on the hilbert space l2 m where a stands for the laplacian on m and h is proportional to the planck constant recall that the quantum flow converges in a sense to the classical flow g in the so called semi classical limit h 0 one can imagine that for small values ofh the quantum system will inherit certain qualitative properties of the classical flow one expects for instance a very different behaviour of eigenfunctions of the laplacian or the distribution of its eigenvalues if the geodesic flow is anosov or in the other extreme completely integrable see sa95 the convergence of the quantum flow to the classical flow is stated in the egorov theorem consider one of the usual quantization procedures op which associates an operator oph a acting on l2 m to every smooth compactly supported function a e c t m on the cotangent bundle t m according to the egorov theorem we have for any fixed t

the sup norm problem on the siegel modular space of rank two let f be a square integrable maa ss form on the siegel upper half space mathcal h of rank 2 for the siegel modular group rm sp 4 bbb z with laplace eigenvalue lambda if in addition f is a joint eigenfunction of the hecke algebra and omega is a compact set in rm sp 4 bbb z backslash mathcal h we show the bound f omega infty ll omega 1 lambda 1 delta for some global constant delta0 as an auxiliary result of independent interest we prove new uniform bounds for spherical functions on semisimple lie groups

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

sieving for mass equidistribution we approach the holomorphic analogue to the quantum unique ergodicity conjecture through an application of the large sieve we deal with shifted convolution sums as in ho109 with various simplifications in our analysis due to the knowledge of the ramanujan petersson conjecture in this holomorphic case

asymptotic rate of quantum ergodicity in chaotic euclidean billiards the quantum unique ergodicity que conjecture of rudnick and sarnak is that every eigenfunction n of the laplacian on a manifold with uniformly hyperbolic geodesic flow becomes equidistributed in the semiclassical limit eigenvalue en that is strong scars are absent we study numerically the rate of equidistribution for a uniformly hyperbolic sinai type planar euclidean billiard with dirichlet boundary condition the drum problem at unprecedented high e and statistical accuracy via the matrix elements n m of a piecewise constant test function a by collecting 30 000 diagonal elements up to level n 7 105 we find that their variance decays with eigenvalue as a power 0 48 0 01 close to the semiclassical estimate of feingold and peres this contrasts with the results of existing studies which have been limited to en a factor 102 smaller we find strong evidence for que in this system we also compare off diagonal variance as a function of distance from the diagonal against feingold peres or spectral measure at the highest accuracy 0 7 thus far in any chaotic system we outline the efficient scaling method and boundary integral formulae used to calculate eigenfunctions 2006 wiley periodicals inc

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

complex zeros of real ergodic eigenfunctions we determine the limit distribution as of complex zeros for holomorphic continuations to grauert tubes of real eigenfunctions of the laplacian on a real analytic compact riemannian manifold m g with ergodic geodesic flow if phi j k is an ergodic sequence of eigenfunctions we prove the weak limit formula frac 1 lambda j z phi j k mathbb c to frac i pi partial bar partial xi g where z phi j k mathbb c is the current of integration over the complex zeros and where overline partial is with respect to the adapted complex structure of lempert szoke and guillemin stenzel

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

scarring on invariant manifolds for perturbed quantized hyperbolic toral automorphisms we exhibit scarring for the quantization of certain nonlinear ergodic maps on the torus we consider perturbations of hyperbolic toral automorphisms preserving certain co isotropic submanifolds the classical dynamics is ergodic hence in the semiclassical limit almost all quantum eigenstates converge to the volume measure of the torus nevertheless we show that for each of the invariant submanifolds there are also eigenstates which localize and converge to the volume measure of the corresponding submanifold

mass equidistribution for hecke eigenforms in this work we study the mass equidistribution for holomorphic hecke eigenforms and establish by employing incomplete poincare series and the petersson formula sharp equidistribution results when the average is performed over intervals much shorter than before a key feature is the analysis of the off diagonal terms that result from this shortening of intervals 2003 wiley periodicals inc

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

equidistribution of eisenstein series on geodesic segments abstract we show an asymptotic formula for the l 2 norm of the eisenstein series e z 1 2 i t restricted to a segment of a geodesic connecting infinity and an arbitrary real for generic geodesics of this form the asymptotic formula shows that the eisenstein series satisfies restricted que on the other hand for rational geodesics the eisenstein series does not satisfy restricted quantum ergodicity as an application we show that the zero set of the eisenstein series intersects every such geodesic segment provided t is large we also make analogous conjectures for the maass cusp forms in particular we predict that cusp forms do not satisfy restricted quantum ergodicity for rational geodesics

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

the unique solution of projective invariants of six points from four uncalibrated images in this paper we show that four uncalibrated images are sufficient to uniquely determine the 3d projective invariants of a set of six points in general position in space an algorithm for computing the unique solution is proposed it computes by solving a set of linear equations for its two linear independent solutions and is simpler than other algorithms which compute the three possible solutions of 3d projective invariants by solving a set of nonlinear equations however a fourth image is needed to get the linear unique solution finally experimental results have shown the feasibility of this algorithm