can billiard eigenstates be approximated by superpositions of plane waves the plane wave decomposition method pwdm is one of the most popular strategies for numerical solution of the quantum billiard problem the method is based on the assumption that each eigenstate in a billiard can be approximated by a superposition of plane waves at a given energy from the classical results on the theory of differential operators this can indeed be justified for billiards in convex domains in contrast in the present work we demonstrate that eigenstates of non convex billiards in general cannot be approximated by any solution of the helmholtz equation regular everywhere in r2 in particular by linear combinations of a finite number of plane waves having the same energy from this we infer that pwdm cannot be applied to billiards in non convex domains furthermore it follows from our results that unlike the properties of integrable billiards where each eigenstate can be extended into the billiard exterior as a regular solution of the helmholtz equation the eigenstates of non convex billiards in general do not admit such an extension

a connection between quantum hilbert space and classical phase space operators we develop a general classicalization procedure that links hilbert space and phase space operators using weyl s operator then we transform the time dependent schrodinger equation into a phase space picture using free parameters they include position q and momentum p we expand the phase space hamiltonian in an taylor series and fix parameters with the condition that coefficients of 0 i1 q and i1 p vanish this condition results in generalized hamilton equations and a natural link between classical and quantum dynamics while the quantum motion equation remains exact in this picture the schrodinger equation reduces in the classical limit to a generalized liouville equation for the quantum mechanical system state we modify glauber s coherent states with a suitable phase factor s q p t and use them to obtain phase space representations of quantum dynamics and quantum mechanical quantities

the inductive blockwise alperin weight condition for g2 q and d43 q abstract the inductive blockwise alperin weight condition is a system of conditions whose verification for all non abelian finite simple groups would imply the blockwise alperin weight conjecture we establish this condition for the groups g 2 q q 5 and d 4 3 q for all primes dividing their order

quantum scars of classical closed orbits in phase space the way in which quantum eigenstates are influenced by the closed orbits of a chaotic classical system is analysed in phase space x q p through the spectral wigner function w x e this is a sum over wigner functions of eigenstates within a range of energy e in the classical limit w is concentrated on the energy surface and smoothly distributed over it closed orbits provide oscillatory corrections scars for which explicit semiclassical formulae are calculated each scar is a fringe pattern decorating the orbit as x moves off the energy surface the fringes form an airy pattern with spacing of order h as x moves off the closed orbit the fringes form a complex gaussian with spacing h

the tate conjecture for k3 surfaces over finite fields artin s conjecture states that supersingular k3 surfaces over finite fields have picard number 22 in this paper we prove artin s conjecture over fields of characteristic p 5 this implies tate s conjecture for k3 surfaces over finite fields of characteristic p 5 our results also yield the tate conjecture for divisors on certain holomorphic symplectic varieties over finite fields with some restrictions on the characteristic as a consequence we prove the tate conjecture for cycles of codimension 2 on cubic fourfolds over finite fields of characteristic p 5

non generic spectral statistics in the quantized stadium billiard we consider the effect of a continuous family of neutral bouncing ball orbits on the energy spectrum of the quantized stadium billiard using a semiclassical approximation we derive analytic expressions for standard two point spectral measures the corrections due to the bouncing ball orbits account for some of the non generic features observed in the analysis of the spectrum of a stadium cavity which was recently measured once the bouncing ball contributions are subtracted the spectrum is shown to be well reproduced by the semi classical trace formula based on unstable periodic orbits we also study special patterns in the spectrum which are due to other non generic features such as edge effects and whispering gallery trajectories

correlations of chaotic eigenfunctions a semiclassical analysis we derive a semiclassical expression for an energy smoothed autocorrelation function defined on a group of eigenstates of the schr odinger equation the system we consider is an energy conserved hamiltonian system possessing time invariant symmetry the energy smoothed autocorrelation function is expressed as a sum of three terms the first one is analogous to berry s conjecture which is a bessel function of the zeroth order the second and the third terms are trace formulae made from special trajectories the second term is found to be direction dependent in the case of spacing averaging which agrees qualitatively with previous numerical observations in high lying eigenstates of a chaotic billiard

survey of the eigenfunctions of a billiard system between integrability and chaos we study numerically the eigenfunctions and their wigner phase space distributions of the two dimensional billiard system defined by the quadratic conformal image of the unit disk as introduced by robnik 1983 this system is a generic kam system and displays a transition from integrability to almost ergodicity as the billiard shape changes we clearly identify two classes of states the regular ones associated with integrable regions and the irregular states supported on classically chaotic regions whilst the mixed type states were not found in support of percival s conjecture 1973 we confirm the existence of extremely intense scars in the classically chaotic regions and demonstrate their association with classical periodic orbits three classes of scars are revealed one orbit scars many orbit one family scars of statistically similar orbits in the homoclinic neighbourhood and many orbit many family scars we argue that it is impossible to find an a priori semiclassical theory of individual eigenstates but do not deny the usefulness of general semiclassical arguments in analysing the collective and statistical properties of eigenstates

statistical analysis of scars in stadium billiard in this paper by using our improved plane wave decomposition method we study the scars in the eigenfunctions of the stadium billiard from a very low state to as high as about the 1 millionth state in the systematic searching for scars of various types we have used the approximate criterion based on the quantization of the classical action along the unstable periodic orbit supporting the scar we have analysed the profile of the integrated probability density along the orbit we found that the maximal integrated intensity of different types of scars scales in different way with the which confirms qualitatively and quantitatively the existing theories of scars such as that of bogomolny and robnik

quantum phase space structures in classically mixed systems a scattering approach the structure of quantum wavefunctions is discussed for a billiard system with mixed classical dynamics using a scattering formalism we introduce a husimi like distribution for s matrix eigenvectors at arbitrary wavenumbers k not necessarily eigenenergies of the system this constitutes probability density plots on the poincare surface of section at any k many eigenvectors are structured by classical objects such as invariant tori the intermediate layer between regular and chaotic motion and unstable periodic orbits scars we find that eigenvector structure is stable over significant energy ranges while the associated eigenphases increase linearly with k at a rate determined by the underlying classical objects additionally we observe that in the presence of time reversal symmetry eigenvectors scarred by non self retracing periodic orbits form doublets which are closely spaced in eigenphase our results provide a new perspective on the quantization of structured states quantization is determined by the linearly increasing eigenphases and occurs whenever an eigenphase is equal to while structure is encoded in the eigenvectors it is present at all energies and varies only slowly this allows one to predict many quantized eigenfunctions over an entire energy range by studying the eigenparameters of at a single intermediate energy and identifying the underlying classical objects

on the number of bouncing ball modes in billiards we study the number of bouncing ball modes in a class of two dimensional quantized billiards with two parallel walls using an adiabatic approximation we show that asymptotically for where depends on the shape of the billiard boundary in particular for the class of two dimensional sinai billiards which are chaotic one can get arbitrarily close from below to which corresponds to the leading term in weyl s law for the mean behaviour of the counting function of eigenstates this result shows that one can come arbitrarily close to violating quantum ergodicity we compare the theoretical results with the numerically determined counting function for the stadium billiard and the cosine billiard and find good agreement

numerical accuracy of bogomolny s semiclassical quantization scheme in quantum billiards we use the semiclassical quantization scheme of bogomolny to calculate eigenvalues of the limacon quantum billiard corresponding to a conformal map of the circle billiard we use the entire billiard boundary as the chosen surface of section and use a finite approximation for the transfer operator in coordinate space computation of the eigenvalues of this matrix combined with a quantization condition determines a set of semiclassical eigenvalues which are compared with those obtained by solving the schrodinger equation the classical dynamics of this billiard system undergoes a smooth transition from integrable circle to completely chaotic motion thus providing a test of bogomolny s semiclassical method in coordinate space in terms of the morphology of the wavefunction we analyse the results for billiards which exhibit both soft and hard chaos

the tate conjecture for k3 surfaces in odd characteristic we show that the classical kuga satake construction gives rise away from characteristic 2 to an open immersion from the moduli of primitively polarized k3 surfaces of any fixed degree to a certain regular integral model for a shimura variety of orthogonal type this allows us to attach to every polarized k3 surface in odd characteristic an abelian variety such that divisors on the surface can be identified with certain endomorphisms of the attached abelian variety in turn this reduces the tate conjecture for k3 surfaces over finitely generated fields of odd characteristic to a version of the tate conjecture for certain endomorphisms on the attached kuga satake abelian variety which we prove as a by product of our methods we also show that the moduli stack of primitively polarized k3 surfaces of degree 2d is quasi projective and when d is not divisible by p 2 is geometrically irreducible in characteristic p we indicate how the same method applies to prove the tate conjecture for co dimension 2 cycles on cubic fourfolds

mapping class group and a global torelli theorem for hyperkahler manifolds a mapping class group of an oriented manifold is a quotient of its diffeomorphism group by the isotopies we compute a mapping class group of a hyperkahler manifold m showing that it is commensurable to an arithmetic lattice in so 3 b2 3 a teichmuller space of m is a space of complex structures on m up to isotopies we define a birational teichmuller space by identifying certain points corresponding to bimeromorphically equivalent manifolds we show that the period map gives the isomorphism between connected components of the birational teichmuller space and the corresponding period space so b2 3 3 so 2 so b2 3 1 we use this result to obtain a torelli theorem identifying each connected component of the birational moduli space with a quotient of a period space by an arithmetic group when m is a hilbert scheme of n points on a k3 surface with n 1 a prime power our torelli theorem implies the usual hodge theoretic birational torelli theorem for other examples of hyperkahler manifolds the hodge theoretic torelli theorem is known to be false

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

smoothed wave functions of chaotic quantum systems abstract it is shown that wave functions of quantum systems as 0 have an extra density near unstable periodic trajectories of the classical problem the averaged wave function square is represented as the sum over a finite number of periodic trajectories the contribution of each trajectory is expressed through the elements of the monodromy matrix of the trajectory the results are compared with the numerical calculations of the wave functions for the stadium billiard

moduli spaces of stable sheaves on abelian surfaces in this paper we consider basic problems on moduli spaces of stable sheaves on abelian surfaces our main assumption is the primitivity of the associated mukai vector we determine the deformation types albanese maps bogomolov factors and their weight 2 hodge structures we also discuss the deformation types of moduli spaces of stable sheaves on k3 surfaces

statistical properties of high lying chaotic eigenstates we study the statistical properties of the high lying chaotic eigenstates 200000 and above which are deep within the semiclassical regime the system we are analysing is the billiard system inside the region defined by the quadratic complex conformal map of the unit disk as introduced by robnik 1983 we are using heller s method of plane wave decomposition of the numerical eigenfunctions and perform extensive statistical analysis with the following conclusions i the local average probability density is in excellent agreement with the microcanonical assumption and all statistical properties are also in excellent agreement with the gaussian random model ii the autocorrelation function is found to be strongly direction dependent and only after averaging over all directions agrees well with berry s 1977 prediction iii although the scars of unstable classical periodic orbits in such an ergodic regime are expected to exist so far we have not found any around the 200000th state other than a scar like feature resembling the whispering gallery modes

how chaotic is the stadium billiard a semiclassical analysis the impression gained from the literature published to date is that the spectrum of the stadium billiard can be adequately described semiclassically by the gutzwiller periodic orbit trace formula together with a modified treatment of the marginally stable family of bouncing ball orbits i show that this belief is erroneous the gutzwiller trace formula is not applicable for the phase space dynamics near the bouncing ball orbits unstable periodic orbits close to the marginally stable family in phase space cannot be treated as isolated stationary phase points when approximating the trace of the green s function semiclassical contributions to the trace show an dependent transition from hard chaos to integrable behaviour for trajectories approaching the bouncing ball orbits a whole region in phase space surrounding the marginal stable family acts semiclassically like a stable island with boundaries being explicitly dependent the localized bouncing ball states found in the billiard derive from this semiclassically stable island the bouncing ball orbits themselves however do not contribute to individual eigenvalues in the spectrum an ebk like quantization of the regular bouncing ball eigenstates in the stadium can be derived the stadium billiard is thus an ideal model for studying the influence of almost regular dynamics near marginally stable boundaries on quantum mechanics the behaviour is generically found at the border of classically stable islands in systems with a mixed phase space structure

wavefunctions expectation values and scars on poincare sections a scattering approach using the scattering approach to quantization we define an appropriate representation of the billiard wavefunction on the scattering poincar section the expectation values of smooth operators in terms of these poincar section wavefunctions are expressed as sums over periodic orbits a special operator is used to define scars on the section and the relation to scars in configuration space is discussed and demonstrated numerically