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

a multiplicity theorem for problems with the p laplacian abstract we consider nonlinear elliptic equations driven by the p laplacian differential operator using degree theoretic arguments based on the degree map for operators of type s we prove the existence of two nontrivial smooth solutions one of which is of constant sign

equipartition of the eigenfunctions of quantized ergodic maps on the torus we give a simple proof of the equipartition of the eigenfunctions of a class of quantized ergodic area preserving maps on the torus examples are the irrational translations the skew translations the hyperbolic automorphisms and some of their perturbations

chaos revealing multiplicative representation of quantum eigenstates the quantisation of the two dimensional toric and spherical phase spaces is considered in analytic coherent state representations every pure quantum state admits therein a finite multiplicative parametrisation by the zeros of its husimi function for eigenstates of quantised systems this description explicitly reflects the nature of the underlying classical dynamics in the semiclassical regime the distribution of the zeros in the phase space becomes one dimensional for integrable systems and highly spread out conceivably uniform for chaotic systems this multiplicative representation thereby acquires a special relevance for semiclassical analysis in chaotic systems

semiclassical asymptotics gauge fields and quantum chaos a gauge field is given by a connection on a principal bundle p m we consider the semiclassical behavior of a family of schrodinger operators associated with a gauge field in the limit as h 0 we relate the spectral theory of such operators to behavior of the hamiltonian flow on the natural phase space associated to a gauge field examining in particular situations where this flow exhibits chaotic behavior

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

exact correlation functions for dual unitary lattice models in 1 1 dimensions we consider a class of quantum lattice models in 1 1 dimensions represented as local quantum circuits that enjoy a particular dual unitarity property in essence this property ensures that both the evolution in time and that in space are given in terms of unitary transfer matrices we show that for this class of circuits generically nonintegrable one can compute explicitly all dynamical correlations of local observables our result is exact nonpertubative and holds for any dimension d of the local hilbert space in the minimal case of qubits d 2 we also present a classification of all dual unitary circuits which allows us to single out a number of distinct classes for the behavior of the dynamical correlations we find noninteracting classes where all correlations are preserved the ergodic and mixing one where all correlations decay and interestingly also classes that are both interacting and nonergodic

quantum ergodicity of boundary values of eigenfunctions suppose that n is a bounded piecewise smooth domain we prove that the boundary values cauchy data of eigenfunctions of the laplacian on with various boundary conditions are quantum ergodic if the classical billiard map on the ball bundle b is ergodic our proof is based on the classical observation that the boundary values of an interior eigenfunction 2 is an eigenfunction of an operator fh on the boundary of with h 1 in the case of the neumann boundary condition fh is the boundary integral operator induced by the double layer potential we show that fh is a semiclassical fourier integral operator quantizing the billiard map plus a small remainder the quantum dynamics defined by fh can be exploited on the boundary much as the quantum dynamics generated by the wave group were exploited in the interior of domains with corners and ergodic billiards in the work of zelditch zworski 1996 novelties include the facts that fh is not unitary and consequently the boundary values are equidistributed by measures which are not invariant under and which depend on the boundary conditions ergodicity of boundary values of eigenfunctions on domains with ergodic billiards was conjectured by s ozawa 1993 and was almost simultaneously proved by gerard leichtnam 1993 in the case of convex c1 1 domains with continuous tangent planes and with dirichlet boundary conditions our methods seem to be quite different motivation to study piecewise smooth domains comes from the fact that almost all known ergodic domains are of this form

existence of a non trivial solution for the p laplacian equation with fucik type resonance at infinity abstract we prove the existence of a non trivial weak solution for the equation p u a u p 1 b u p 1 f x u in u 0 on by variational methods we consider the case where a 1 or b 1 1 denotes the first eigenvalue of p or the case where a is close to b while the nonlinearity f belongs to a class that contains for example u q 2 u 1 q p

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

multiple solutions for elliptic problems with asymmetric nonlinearity in this paper we establish the existence of multiple solutions for the semilinear elliptic problem u g x u in u 0on where rn is an open bounded domain with smooth boundary g r r is a function of class c1 such that g x 0 0 and which is asymptotically linear at infinity with jumping nonlinearities we use critical groups to distinguish the critical points

ergodicity of eigenfunctions for ergodic billiards we give a simple proof of ergodicity of eigenfunctions of the laplacian with dirichlet boundary conditions on compact riemannian manifolds with piecewise smooth boundaries and ergodic billiards examples include the bunimovich stadium the sinai billiard and the generic polygonal billiard tables of kerckhoff masur and smillie

existence of changing sign solutions for some semilinear problems with jumping nonlinearities at zero we study the existence of changing sign solutions of an elliptic semilinear boundary value problem which arises as a limiting equation of the two species lotka volterra competing equations system using variational methods and a result of d aujourd hui we find conditions which are both sufficient and necessary for this existence problem

ergodicite et limite semi classique consider a self adjoint quantic hamiltonian p h p x hdx whereh 0 is the planck s constant andp some smooth classical observable on the phase space r2n when the classical flow on a compact energy shell p is ergodic we prove that in the limith 0 almost all the eigenfunctions ofp h whose energy is near of are distributed according to the liouville measure on p

on the critical percolation probabilities we prove that the critical probabilities of site percolation on the square lattice satisfy the relation p c p 1 furthermore we prove the continuity of the function percolation probability

a mating of trees approach for graph distances in random planar maps we introduce a general technique for proving estimates for certain random planar maps which belong to the gamma liouville quantum gravity lqg universality class for gamma in 0 2 the family of random planar maps we consider are those which can be encoded by a two dimensional random walk with i i d increments via a mating of trees bijection and includes the uniform infinite planar triangulation uipt gamma sqrt 8 3 and planar maps weighted by the number of different spanning trees gamma sqrt 2 bipolar orientations gamma sqrt 4 3 or schnyder woods gamma 1 that can be put on the map using our technique we prove estimates for graph distances in the above family of random planar maps in particular we obtain non trivial upper and lower bounds for the cardinality of a graph distance ball consistent with the watabiki prog theor phys suppl 114 1 17 1993 prediction for the hausdorff dimension of gamma lqg and we establish the existence of an exponent for certain distances in the map the basic idea of our approach is to compare a given random planar map m to a mated crt map a random planar map constructed from a correlated two dimensional brownian motion using a strong coupling zaitsev in esaim probab stat 2 41 108 1998 of the encoding walk for m and the brownian motion used to construct the mated crt map this allows us to deduce estimates for graph distances in m from the estimates for graph distances in the mated crt map which we proved using continuum theory in a previous work in the special case when gamma sqrt 8 3 we instead deduce estimates for the sqrt 8 3 mated crt map from known results for the uipt the arguments of this paper do not directly use sle lqg and can be read without any knowledge of these objects

high energy limits of laplace type and dirac type eigenfunctions and frame flows we relate high energy limits of laplace type and dirac type operators to frame flows on the corresponding manifolds and show that the ergodicity of frame flows implies quantum ergodicity in an appropriate sense for those operators observables for the corresponding quantum systems are matrix valued pseudodifferential operators and therefore the system remains non commutative in the high energy limit we discuss to what extent the space of stationary high energy states behaves classically

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

a brezis nirenberg splitting approach for nonlocal fractional equations in this paper we consider problems modeled by the following nonlocal fractional equation su a x u f u in u 0in rn where s 0 1 is fixed is an open bounded subset of rn n 2s with lipschitz boundary s is the fractional laplace operator and is a real parameter under two different types of conditions on the functions a and f by using a famous critical point theorem in the presence of splitting established by brezis and nirenberg we obtain the existence of at least two nontrivial weak solutions for our problem

geodesics in large planar maps and in the brownian map we study geodesics in the random metric space called the brownian map which appears as the scaling limit of large planar maps in particular we completely describe geodesics starting from the distinguished point called the root and we characterize the set s of all points that are connected to the root by more than one geodesic the set s is dense in the brownian map and homeomorphic to a non compact real tree furthermore for every x in s the number of distinct geodesics from x to the root is equal to the number of connected components of s x in particular points of the brownian map can be connected to the root by at most three distinct geodesics our results have applications to the behavior of geodesics in large planar maps

l p norms nodal sets and quantum ergodicity abstract for small range of p 2 we improve the l p bounds of eigenfunctions of the laplacian on negatively curved manifolds our improvement is by a power of logarithm for a full density sequence of eigenfunctions we also derive improvements on the size of the nodal sets our proof is based on a quantum ergodicity property of independent interest which holds for families of symbols supported in balls whose radius shrinks at a logarithmic rate

subsequential scaling limits for liouville graph distance for 0 gamma 2 and delta 0 we consider the liouville graph distance which is the minimal number of euclidean balls of gamma liouville quantum gravity measure at most delta whose union contains a continuous path between two endpoints in this paper we show that the renormalized distance is tight and thus has subsequential scaling limits at delta rightarrow 0 in particular we show that for all delta 0 the diameter with respect to the liouville graph distance has the same order as the typical distance between two endpoints

no quantum ergodicity for star graphs we investigate statistical properties of the eigenfunctions of the schrodinger operator on families of star graphs with incommensurate bond lengths we show that these eigenfunctions are not quantum ergodic in the limit as the number of bonds tends to infinity by finding an observable for which the quantum matrix elements do not converge to the classical average we further show that for a given fixed graph there are subsequences of eigenfunctions which localise on pairs of bonds we describe how to construct such subsequences explicitly these structures are analogous to scars on short unstable periodic orbits

on the rate of quantum ergodicity on hyperbolic surfaces and for billiards abstract the rate of quantum ergodicity is studied for three strongly chaotic anosov systems the quantal eigenfunctions on a compact riemannian surface of genus g 2 and of two triangular billiards on a surface of constant negative curvature are investigated one of the triangular billiards belongs to the class of arithmetic systems there are no peculiarities observed in the arithmetic systems concerning the rate of quantum ergodicity this contrasts to the peculiar behaviour with respect to the statistical properties of the quantal levels it is demonstrated that the rate of quantum ergodicity in the three considered systems fits well with the known upper and lower bounds furthermore sarnak s conjecture about quantum unique ergodicity for hyperbolic surfaces is confirmed numerically in these three systems

continuum quantum systems as limits of discrete quantum systems i state vectors abstract dynamical systems on continuum hilbert spaces may be realized as limits of dynamical systems on discrete possibly finite dimensional hilbert spaces in this first of four papers on the topic the continuum and discrete spaces are interfaced to one another algebraically convergence of vectors is defined in such a way as to preserve inner products and a necessary and sufficient coordinate wise criterion for convergence is proved

quantized kronecker flows and almost periodic quantum field theory we define and study the properties of the infinite dimensional quantized kronecker flow this c dynamical system arises as a quantization of the corresponding flow on an infinite dimensional torus we prove an ergodic theorem for a class of quantized kronecker flows we also study the closely related almost periodic quantum field theory of bosonic fermionic and supersymmetric particles we prove the existence and uniqueness of kms and super kms states for the c algebras of observables arising in these theories

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

scaling exponents and multifractal dimensions for independent random cascades this paper is concerned with mandelbrot s stochastic cascade measures the problems of i scaling exponents of structure functions of the measure q and ii multifractal dimensions are considered for cascades with a generator vector w1 wc of the general type these problems were previously studied for independent strongly bounded variableswi 0 a wi c consequently a broad class of models used in applications including kolmogorov s long normal model in turbulence long stable universal cascades in atmospheric dynamics has not been covered roughly speaking problems i ii are here solved under the condition that the scaling exists the function is calculated for all arguments previously this was done for positiveq and a new effect emerges the function can generally involve discontinuities in the first derivative as well as in the second

intertwining the geodesic flow and the schrodinger group on hyperbolic surfaces we construct an explicit intertwining operator mathcal l between the schrodinger group e it frac triangle2 and the geodesic flow on certain hilbert spaces of symbols on the cotangent bundle t x of a compact hyperbolic surface x d we also define invariant eigendistributions of the geodesic flow ps j k nu j nu k patterson sullivan distributions out of pairs of triangle eigenfunctions generalizing the diagonal case j k treated in anantharaman and zelditch ann henri poincare 8 2 361 426 2007 the operator mathcal l maps ps j k nu j nu k to the wigner distribution w gamma j k studied in quantum chaos we define hilbert spaces mathcal h ps whose dual is spanned by ps j k nu j nu k resp mathcal h w whose dual is spanned by w gamma j k and show that mathcal l is a unitary isomorphism from mathcal h w to mathcal h ps

critical points and positive solutions of singular elliptic boundary value problems usually we do not think there is variational structure for singular elliptic boundary value problems so it cannot be considered by using critical points theory in this paper we use critical theory on certain convex closed sets to solve positive solutions for singular elliptic boundary value problems especially use the ordinary differential equation theory of banach spaces to obtain new results on the existence of multiple positive solutions the method is useful for other singular problems

a distance exponent for liouville quantum gravity let gamma in 0 2 and let h be the random distribution on mathbb c which describes a gamma liouville quantum gravity lqg cone also let kappa 16 gamma 2 4 and let eta be a whole plane space filling sle kappa curve sampled independent from h and parametrized by gamma quantum mass with respect to h we study a family mathcal g epsilon epsilon 0 of planar maps associated with h eta called the lqg structure graphs a k a mated crt maps which we conjecture converge in probability in the scaling limit with respect to the gromov hausdorff topology to a random metric space associated with gamma lqg in particular mathcal g epsilon is the graph whose vertex set is epsilon mathbb z with two such vertices x 1 x 2 in epsilon mathbb z connected by an edge if and only if the corresponding curve segments eta x 1 epsilon x 1 and eta x 2 epsilon x 2 share a non trivial boundary arc due to the peanosphere description of sle decorated lqg due to duplantier et al liouville quantum gravity as a mating of trees 2014 the graph mathcal g epsilon can equivalently be expressed as an explicit functional of a correlated two dimensional brownian motion so can be studied without any reference to sle or lqg we prove non trivial upper and lower bounds for the cardinality of a graph distance ball of radius n in mathcal g epsilon which are consistent with the prediction of watabiki prog theor phys suppl 114 1 17 1993 for the hausdorff dimension of lqg using subadditivity arguments we also prove that there is an exponent chi 0 for which the expected graph distance between generic points in the subgraph of mathcal g epsilon corresponding to the segment eta 0 1 is of order epsilon chi o epsilon 1 and this distance is extremely unlikely to be larger than epsilon chi o epsilon 1

on bifurcation critical groups and exact multiplicity of solutions to semilinear elliptic boundary value problems in this paper we use a bifurcation approach morse theory to study asymptotically linear and superlinear elliptic boundary value problems and obtain new results on the exact multiplicity of solutions and sign changing solutions to asymptotically linear and superlinear problems and the properties of the solutions

on the distribution of matrix elements for the quantum cat map for many classically chaotic systems it is believed that the quantum wave functions become uniformly distributed that is the matrix elements of smooth observables tend to the phase space average of the observable in this paper we study the fluctuations of the matrix elements for the desymmetrized quantum cat map we present a conjecture for the distribution of the normalized matrix elements namely that their distribution is that of a certain weighted sum of traces of independent matrices in su 2 this is in contrast to generic chaotic systems where the distribution is expected to be gaussian we compute the second and fourth moment of the normalized matrix elements and obtain agreement with our conjecture

multiple solutions of impulsive sturm liouville boundary value problem via lower and upper solutions and variational methods abstract in this paper we prove the existence of multiple solutions for second order sturm liouville boundary value problem l u f x u x 0 1 r 1 u 0 r 2 u 0 where l u p x u q x u is a sturm liouville operator r 1 u u 0 u 0 r 2 u u 1 u 1 the technical approach is fully based on lower and upper solutions and variational methods the interesting point is that the existence of four solutions and seven solutions is given

the semiclassical theory of discontinuous systems and ray splitting billiards we analyze the semiclassical limit of spectral theory on manifolds whose metrics have jump like discontinuities such systems are quite different from manifolds with smooth riemannian metrics because the semiclassical limit does not relate to a classical flow but rather to branching ray splitting billiard dynamics in order to describe this system we introduce a dynamical system on the space of functions on phase space to identify the quantum dynamics in the semiclassical limit we compute the principal symbols of the fourier integral operators associated to reflected and refracted geodesic rays and identify the relation between classical and quantum dynamics in particular we prove a quantum ergodicity theorem for discontinuous systems in order to do this we introduce a new notion of ergodicity for the ray splitting dynamics

chaotic eigenfunctions in phase space we study individual eigenstates of quantized area preserving maps on the 2 torus which are classically chaotic in order to analyze their semiclassical behavior we use the bargmann husimi representations for quantum states as well as their stellar parametrization which encodes states through a minimal set of points in phase space the constellation of zeros of the husimi density we rigorously prove that a semiclassical uniform distribution of husimi densities on the torus entails a similar equidistribution for the corresponding constellations we deduce from this property a universal behavior for the phase patterns of chaotic bargmann eigenfunctions which is reminiscent of the wkb approximation for eigenstates of integrable systems though in a weaker sense in order to obtain more precise information on chaotic eigenconstellations we then model their properties by ensembles of random states generalizing former results on the 2 sphere to the torus geometry this approach yields statistical predictions for the constellations which fit quite well the chaotic data we finally observe that specific dynamical information e g the presence of high peaks like scars in husimi densities can be recovered from the knowledge of a few long wavelength fourier coefficients which therefore appear as valuable order parameters at the level of individual chaotic eigenfunctions

dancer fucik spectrum for fractional schrodinger operators with a steep potential well on rn abstract in this paper we study dancer fucik spectrum of the fractional schrodinger operators which is defined as the set of r 2 such that s u v x u u u in r n has a nontrivial solution u where the potential v has a steep potential well for sufficiently large parameter 0 it is allowed that s v has essential spectrum with finitely many eigenvalues below the infimum of e s s s v many difficulties are caused by general nonlocal operators we develop new techniques to overcome them to construct the first nontrivial curve of dancer fucik point spectrum by minimax methods to show some qualitative properties of the curve and to prove that the corresponding eigenfunctions are foliated schwartz symmetric as applications we obtain the existence of nontrivial solutions for nonlinear schrodinger equations with nonresonant nonlinearity

ergodic properties of quantized toral automorphisms we study the ergodic properties for a class of quantized toral automorphisms namely the cat and kronecker maps the present work uses and extends the results of klimek and le niewski ann phys 244 173 198 1996 we show that quantized cat maps are strongly mixing while kronecker maps are ergodic and nonmixing we also study the structure of these quantum maps and show that they are effected by unitary endomorphisms of a suitable vector bundle over a torus this allows us to exhibit explicit relations between our toeplitz quantization and the semiclassical quantization of cat maps proposed by hannay and berry physica d 1 267 290 1980

weak quantum ergodicity abstract we examine the consequences of classical ergodicity for the localization properties of individual quantum eigenstates in the classical limit we note that the well known schnirelman result is a weaker form of quantum ergodicity than the one implied by random matrix theory this suggests the possibility of systems with non gaussian random eigenstates which are nonetheless ergodic in the sense of schnirelman and lead to ergodic transport in the classical limit these we call weakly quantum ergodic indeed for a class of slow ergodic classical systems it is found that each eigenstate becomes localized to an ever decreasing fraction of the available state space in the semiclassical limit nevertheless each eigenstate in this limit covers phase space evenly on any classical scale and long time transport properties between individual quantum states remain ergodic due to the diffractive effects which dominate quantum phase space exploration

crossing probabilities for voronoi percolation we prove that the standard russo seymour welsh theory is valid for voronoi percolation this implies that at criticality the crossing probabilities for rectangles are bounded by constants depending only on their aspect ratio this result has many consequences such as the polynomial decay of the one arm event at criticality

classical limit of the quantized hyperbolic toral automorphisms the canonical quantization of any hyperbolic symplectomorphisma of the 2 torus yields a periodic unitary operator on an dimenional hilbert space n 1 h we prove that this quantum system becomes ergodic and mixing at the classical limit n n prime which can be interchanged with the time average limit the recovery of the stochastic behaviour out of a periodic one is based on the same mechanism under which the uniform distribution of the classical periodic orbits reproduces the lebesgue measure the wigner functions of the eigenstates supported on the classical periodic orbits are indeed proved to become uniformly speread in phase space

on the entropy of quantum limits for 2 dimensional cat maps we study semiclassical measures or quantum limits for quantized hyperbolic automorphisms of mathbb t 2 we show that any quantum limit has the following property if a weight is carried on ergodic components of low entropy say entropy less than h0 then a weight must be carried on ergodic components of high entropy hmax h0 where hmax is the maximal entropy this combines some existing partial results towards the classification of quantum limits

residue traces for certain algebras of fourier integral operators abstract in this paper a trace functional is defined for certain algebras of fourier integral operators including for instance the algebra of toeplitz operators on the boundary of a strictly pseudoconvex domain and it is shown that vanishing of this trace characterizes commutators in each of these algebras

uniqueness and universality of the brownian map we consider a random planar map mn which is uniformly distributed over the class of all rooted q angulations with n faces we let mn be the vertex set of mn which is equipped with the graph distance dgr both when q 4 is an even integer and when q 3 there exists a positive constant cq such that the rescaled metric spaces mn cqn 1 4dgr converge in distribution in the gromov hausdorff sense toward a universal limit called the brownian map the particular case of triangulations solves a question of schramm

moyal quantization with compact symmetry groups and noncommutative harmonic analysis the phase space approach to quantization of systems whose symmetry group is compact and semisimple is developed from two basic principles covariance and traciality this generalizes results and methods already implemented for spin systems j c varilly and j m gracia bondia ann phys 190 107 1989 the twisted product of phase space functions is shown to be the image of group convolution in the context of a novel fourier theory on the coadjoint orbits

on sign changing and multiple solutions of the p laplacian abstract in this paper we construct the pseudo gradient vector field in w 0 1 p by which we obtain the positive and negative cones of w 0 1 p are both invariant sets of the descent flow of the corresponding functional then we use differential equations theory in banach spaces and dynamics theory to study p laplacian boundary value problems with jumping nonlinearities at zero or infinity and get new multiple solutions and sign changing solutions theorems of p laplacian

non universality for first passage percolation on the exponential of log correlated gaussian fields we consider first passage percolation fpp where the vertex weight is given by the exponential of two dimensional log correlated gaussian fields our work is motivated by understanding the discrete analog for the random metric associated with liouville quantum gravity lqg which roughly corresponds to the exponential of a two dimensional gaussian free field gff the particular focus of the present paper is an aspect of universality for such fpp among the family of log correlated gaussian fields more precisely we construct a family of log correlated gaussian fields and show that the fpp distance between two typically sampled vertices according to the lqg measure is n 1 o varepsilon where n is the side length of the box and varepsilon can be made arbitrarily small if we tune a certain parameter in our construction that is the exponents can be arbitrarily close to 1 combined with a recent work of the first author and goswami on an upper bound for this exponent when the underlying field is a gff our result implies that such exponent is not universal among the family of log correlated gaussian fields

the brownian map is the scaling limit of uniform random plane quadrangulations we prove that uniform random quadrangulations of the sphere with n faces endowed with the usual graph distance and renormalized by n 1 4 converge as n in distribution for the gromov hausdorff topology to a limiting metric space we validate a conjecture by le gall by showing that the limit is up to a scale constant the so called brownian map which was introduced by marckert mokkadem and le gall as the most natural candidate for the scaling limit of many models of random plane maps the proof relies strongly on the concept of geodesic stars in the map which are configurations made of several geodesics that only share a common endpoint and do not meet elsewhere

construction of pseudo gradient vector field and sign changing multiple solutions involving p laplacian existence of multiple and sign changing solutions for a problem involving p laplacian and jumping nonlinearities are considered via the construction of descent flow in c01 sign changing and multiple solutions are obtained under additional assumption on the nonlinearity the uniqueness of positive negative solution theorem is included too

non uniformly asymptotically linear p laplacian problems abstract we relax conditions on uniform convergences of f x s s p 2 s which are essential in many results on multiplicity for asymptotically linear p laplacian problems

fucik spectrum sign changing and multiple solutions for semilinear elliptic boundary value problems with resonance at infinity in this paper we use fucik spectrum ordinary differential equation theory of banach spaces to study semilinear elliptic boundary value problems with jumping nonlinearities at zero or infinity especially with resonance at infinity and obtain new results on the existence of multiple solutions and sign changing solutions with a weakening of the p s condition in one case we get up to seven nontrivial solutions the techniques have independent interest

on a quantum chaos theorem of r schrader and m taylor abstract we extend results of r schrader and m taylor on the semi classical asymptotics of eigenfunctions of a quantum hamiltonian for a particle in a gauge field our method is based on the guillemin sternberg uribe theory of fuzzy ladders and reductions of hamiltonian systems with symmetry

a semiclassical egorov theorem and quantum ergodicity for matrix valued operators we study the semiclassical time evolution of observables given by matrix valued pseudodifferential operators and construct a decomposition of the hilbert space l2 open image in new windowd open image in new windown into a finite number of almost invariant subspaces for a certain class of observables that is preserved by the time evolution we prove an egorov theorem we then associate with each almost invariant subspace of l2 open image in new windowd open image in new windown a classical system on a product phase space t open image in new windowd open image in new window where open image in new window is a compact symplectic manifold on which the classical counterpart of the matrix degrees of freedom is represented for the projections of eigenvectors of the quantum hamiltonian to the almost invariant subspaces we finally prove quantum ergodicity to hold if the associated classical systems are ergodic

analytic representations in quantum mechanics various euclidean hyperbolic and elliptic analytic representations are introduced and relations among them are discussed the bargmann analytic representation in the complex plane is considered and its relation to other phase space methods for the harmonic oscillator is reviewed the general theory that relates the growth of analytic functions with the density of their zeros is applied to bargmann functions and it leads to theorems on the completeness of sequences of glauber coherent states two hyperbolic analytic representations in the unit disc based on su 1 1 coherent states and also on phase states are introduced a third analytic representation in the complex plane based on barut girardello states is also considered and transformations which relate it to the other ones are studied in the case of systems with finite dimensional hilbert space an elliptic analytic representation in the extended complex plane and also another analytic representation based on theta functions are introduced the berezin formalism in the euclidean hyperbolic and elliptic cases is discussed contour analytic representations in these three cases are also presented

on quantum ergodicity for linear maps of the torus we prove a strong version of quantum ergodicity for linear hyperbolic maps of the torus cat maps we show that there is a density one sequence of integers so that as n tends to infinity along this sequence all eigenfunctions of the quantum propagator at inverse planck constant n are uniformly distributed

sign changing solutions of kirchhoff type problems via invariant sets of descent flow we obtain sign changing solutions of a class of nonlocal quasilinear elliptic boundary value problems using variational methods and invariant sets of descent flow

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

lognormal star scale invariant random measures in this article we consider the continuous analog of the celebrated mandelbrot star equation with lognormal weights mandelbrot introduced this equation to characterize the law of multiplicative cascades we show existence and uniqueness of measures satisfying the aforementioned continuous equation these measures fall under the scope of the gaussian multiplicative chaos theory developed by kahane in 1985 or possibly extensions of this theory as a by product we also obtain an explicit characterization of the covariance structure of these measures we also prove that qualitative properties such as long range independence or isotropy can be read off the equation

sign changing and multiple solutions of kirchhoff type problems without the p s condition abstract the existence of nontrivial solutions of kirchhoff type equations is an important nonlocal quasilinear problem in this paper we use minimax methods and invariant sets of descent flow to prove two interesting existence theorems for the following 4 superlinear kirchhoff type problems without the p s condition one concerning the existence of a nontrivial solution and the other one concerning the existence of sign changing solutions and multiple solutions a b u 2 u f x u in u 0 on

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

disheveled arnold s cat and the problem of quantum classic correspondence abstract quantum arnold s cat map is studied for a case of perfect square inverse planck s constant n m 2 the classic limit is analyzed on a subset of numbers n increasing as 4 k the quantum problem in this case allows exact reduction to the classic cat map defined on a discrete lattice of size m m and supplemented by evolution of a phase variable a link between the classic periodic orbits and spectrum of eigenvalues of the quantum evolution operator is outlined for m growing as 2 k genetic analysis is developed for periodic orbits and they are classified by means of a tree like graph a phase shift accumulated over a period of the orbits evolves from level to level of the graph according to a certain rule governed by non periodic binary code representation of a localized gaussian wave packet in a basis of eigenvectors of the evolution operator gives rise to a probability measure distributed on a unit circle where the eigenvalues are located this measure looks like spectrum of a finite time sample of a stationary random process periodogram 1 majority of the eigenstates have intensities of comparable order of magnitude 2 the spectral distribution is of locally random like nature i e statistical variance of the amplitudes has the same order as the amplitudes themselves this combination of properties in very straightforward manner follows from chaotic nature of the classic map and is conjectured to be the most fundamental attribute of quantum chaos

multiple and nodal solutions for parametric neumann problems with nonhomogeneous differential operator and critical growth abstract we consider a parametric neumann problem with nonhomogeneous differential operator and critical growth combining variational methods based on critical point theory with suitable truncation techniques and flow invariance arguments we show that for all large the problem has at least three nontrivial smooth solutions two of constant sign one positive the other negative and the third nodal we also study the asymptotic behavior of all solutions obtained when converges to infinity the interesting point is that we do not impose any restrictions to the behavior of the nonlinear term f at infinity our work unifies and sharply improves several recent papers

sign changing solutions for resonant neumann problems we consider semilinear neumann problems at resonance we are concerned with two distinct cases in the first one the potential function is unbounded and indefinite in the second case the potential function is bounded and the reaction exhibits a concave parametric term near the origin we prove two existence theorems producing sign changing solutions the first main result is a complement of the results obtained by papageorgiou and r dulescu 2015 22 the second theorem gives an answer to the open question raised by candito et al 2016 4 our approach uses variational methods together with flow invariance arguments

oscillator eigenstates concentrated on classical trajectories the highly degenerate eigenspaces of the two dimensional isotropic harmonic oscillator are proven to contain eigenstates that are optimally localized on the closed trajectories of the classical dynamics as h cross to 0 their phase space probability density converges to the unique probability density on the corresponding trajectory which is invariant under the classical flow

quantization of a class of piecewise affine transformations on the torus we present a unified framework for the quantization of a family of discrete dynamical systems of varying degrees of chaoticity the systems to be quantized are piecewise affine maps on the two torus viewed as phase space and include the automorphisms translations and skew translations we then treat some discontinuous transformations such as the baker map and the sawtooth like maps our approach extends some ideas from geometric quantization and it is both conceptually and calculationally simple

spectra of phase point operators in odd prime dimensions and the extended clifford group we analyze the role of the extended clifford group in classifying the spectra of phase point operators within the framework laid out by gibbons et al phys rev a 70 062101 2004 for setting up wigner distributions on discrete phase spaces based on finite fields to do so we regard the set of all the discrete phase spaces as a symplectic vector space over the finite field auxiliary results include a derivation of the conjugacy classes of esl 2 fn

universality in the flooding of regular islands by chaotic states we investigate the structure of eigenstates in systems with a mixed phase space in terms of their projection onto individual regular tori depending on dynamical tunneling rates and the heisenberg time regular states disappear and chaotic states flood the regular tori for a quantitative understanding we introduce a random matrix model the resulting statistical properties of eigenstates as a function of an effective coupling strength are in very good agreement with numerical results for a kicked system we discuss the implications of these results for the applicability of the semiclassical eigenfunction hypothesis

sign changing and multiple solutions of the sturm liouville boundary value problem via invariant sets of descending flow abstract in this paper we prove the existence of sign changing and multiple solutions for the second order sturm liouville boundary value problem l u f x u x 0 1 r 1 u 0 r 2 u 0 where l u p x u q x u is the sturm liouville operator r 1 u u 0 u 0 and r 2 u u 1 u 1 the technical approach is fully based on minimax methods and invariant sets of descending flow

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

sign changing solutions and multiple solutions theorems for semilinear elliptic boundary value problems with a reaction term nonzero at zero abstract in this paper we use the ordinary differential equation theory of banach spaces and minimax theory in particular the local mountain pass lemma to study asymptotically linear and superlinear elliptic boundary value problems with a reaction term nonzero at zero and get new multiple solutions and sign changing solutions theorems moreover the sign changing solutions change sign exactly once under some conditions

on the existence of signed and sign changing solutions for a class of superlinear schrodinger equations this paper deals with a semilinear schrodinger equation whose nonlinear term involves a positive parameter and a real function f u which satisfies a superlinear growth condition just in a neighborhood of zero by proving an a priori estimate for a suitable class of solutions we are able to avoid further restrictions on the behavior of f u at infinity in order to prove for sufficiently large the existence of one sign and sign changing solutions minimax methods are employed to establish this result