dissipation in wigner poisson systems the wigner poisson wp system or quantum vlasov poisson system is modified to include dissipative terms in the hamiltonian by utilizing the equivalence of the wp system to the schrodinger poisson system global existence and uniqueness are proved and regularity properties are deduced the proof differs somewhat from that for the nondissipative case treated previously by brezzi markowich and illner et al in particular the hille yosida theorem is used since the linear evolution is not unitary and a liapunov function is introduced to replace the energy which is not conserved

ground state solution of nehari pohožaev type for periodic quasilinear schrodinger system this paper is concerned with a quasilinear schrodinger system by minimization under a convenient constraint and concentration compactness lemma we prove the existence of ground state solutions our result covers the case of 2 4 which seems to be the first result for a coupled quasilinear schrodinger system in the periodic situation

stationary solutions of quasi linear schrodinger poisson systems abstract we show that a high field version of the periodic schrodinger poisson system including nonlinear terms in the poisson equation corresponding to a field dependent dielectric constant and effective potentials in the schrodinger equation has an infinite number of different stationary states which correspond to solutions of a nonlinear schrodinger poisson eigenvalue problem

sign changing solutions for quasilinear schrodinger equations with restraint abstract we study the existence of sign changing solutions for the quasilinear schrodinger equation u v x u 1 2 u u 2 u p 1 u in r n with the restraint r n u p 1 d x 1 where appears as a lagrangian multiplier 4 p 1 4 n n 2 and v v x x r n is a given potential

an operator splitting method for the wigner poisson problem the wigner poisson equation describes the quantum mechanical motion of electrons in a self consistent electrostatic field the equation consists of a transport term and a non linear pseudodifferential operator in this paper we analyze an operator splitting method for the linear wigner equation and the coupled wigner poisson problem for this semidiscretization in time consistency and nonlinear stability are established in an l2 framework we present a numerical example to illustrate the method

existence of soliton solutions for a quasilinear schrodinger equation involving critical exponent in rn abstract mountain pass in a suitable orlicz space is employed to prove the existence of soliton solutions for a quasilinear schrodinger equation involving critical exponent in r n these equations contain strongly singular nonlinearities which include derivatives of the second order such equations have been studied as models of several physical phenomena the nonlinearity here corresponds to the superfluid film equation in plasma physics

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

some existence results of solutions for the semilinear elliptic equations on rn abstract in this paper we prove existence of w 1 2 r n solutions for the following equations on r n u q x u u g x u where the operator q x is indefinite and g x z is superlinear or asymptotically linear as z

the metric entropy of diffeomorphisms part ii relations between entropy exponents and dimension on considere f m m m m un c 2 diffeomorphisme f d une variete de riemann compacte m preservant une mesure de probabilite de borel m soit hm f l entropie metrique de f et 1 x r x x les exposants de lyapunov distincts en x on etablit une formule qui relie h m f et les r x x

multiple solutions of schrodinger equations with indefinite linear part and super or asymptotically linear terms abstract based on new information concerning strongly indefinite functionals without palais smale conditions we study existence and multiplicity of solutions of the schrodinger equation u v x u g x u for x r n u x 0 as x where v and g are periodic with respect to x and 0 lies in a gap of v supposing g is asymptotically linear as u and symmetric in u we obtain infinitely many geometrically distinct solutions we also consider the situation where g is super linear with mild assumptions different from those studied previously and establish the existence and multiplicity

local global principles for representations of quadratic forms we prove a local global principle for the problem of representations of quadratic forms by quadratic forms over in codimension 5 the proof uses the ergodic theory of p adic groups together with a fairly general observation on the structure of orbits of an arithmetic group acting on integral points of a variety

multiple solutions for a class of nonlinear schrodinger equations abstract in this paper we find new conditions to ensure the existence of infinitely many homoclinic type solutions for the schrodinger equation u v x u g x u for x r n assuming v x and g x u depend periodically on x we deal with the situations where g x u is as u asymptotically linear or superlinear with certain hypothesis different from ones used in previous related study our approach is variational and we use the cerami condition instead of the palais smale one for deformation arguments

from rational billiards to dynamics on moduli spaces this short expository note gives an elementary introduction to the study of dynamics on certain moduli spaces and in particular the recent breakthrough result of eskin mirzakhani and mohammadi we also discuss the context and applications of this result and connections to other areas of mathematics such as algebraic geometry teichmuller theory and ergodic theory on

the metric entropy of diffeomorphisms part i characterization of measures satisfying pesin s entropy formula soit m une variete de riemann compacte soit f m m un diffeomorphisme et soit m une mesure de probabilite de borel f invariante sur m on identifie les mesures pour lesquelles l inegalite de margulis et ruelle qui relie l entropie aux exposants de lyapunov atteint l egalite on demontre une formule valable pour toutes les mesures invariantes

on a problem in simultaneous diophantine approximation littlewood s conjecture a consequence of hurwitz s theorem is that the right hand side of the above inequality cannot be improved by an arbitrary positive constant s more precisely for s l x 5 there exist real numbers a e i for which the inequality ilqall cq 1 has at most a finite number of solutions these a are the badly approximable numbers and we will denote by b a d the set of all such numbers that is

on raghunathan s measure conjecture this paper represents the last in our three part series on raghunathan s measure conjecture see r1 r2 for parts i and ii more specifically let g be a real lie group all groups in this paper are assumed to be second countable with the lie algebra r a discrete subgroup of g and ir g r g the projection lwg rg the group g acts by right translations on r g x x g x e r g g e g let gt be a borel probability measure on r g define

time dependent dissipation in nonlinear schrodinger systems a coupled nonlinear schrodinger poisson equation is considered which contains a time dependent dissipation function as a specific model of dissipation effects in nonlinear quantum transport theory and other areas the wigner poisson equation associated with this system is derived using conservation and quasiconservation laws and certain growth assumptions for the nonlinearities and the dissipation function global existence of solutions to the cauchy problem of the time dependent schrodinger poisson system is shown both for small attractive case or arbitrary data repulsive case

invariant measures on g γ for split simple lie groups g we study the left action of a cartan subgroup on the space x g where is a lattice in a simple split connected lie group g of rank n 1 let be an invariant measure on x we give several conditions using entropy and conditional measures each of which characterizes the haar measure on x furthermore we show that the conditional measure on the foliation of unstable manifolds has the structure of a product measure the main new element compared to the previous work on this subject is the use of noncommutativity of root foliations to establish rigidity of invariant measures 2003 wiley periodicals inc

soliton solutions for quasilinear schrodinger equations ii abstract for a class of quasilinear schrodinger equations we establish the existence of ground states of soliton type solutions by a variational method

on a class of periodic quasilinear schrodinger equations involving critical growth in r2 abstract we consider the equation u v x u k 2 u 2 u g x u u 0 x r 2 where v r 2 r and g r 2 r r are two continuous 1 periodic functions and k is a positive constant also we assume g behaves like exp u 4 as u we prove the existence of at least one weak solution u h 1 r 2 with u 2 h 1 r 2 the mountain pass in a suitable orlicz space together with the trudinger moser inequality are employed to establish this result such equations arise when one seeks for standing wave solutions for the corresponding quasilinear schrodinger equations schrodinger equations of this type have been studied as models of several physical phenomena the nonlinearity here corresponds to the superfluid film equation in plasma physics

invariant measures and the set of exceptions to littlewood s conjecture we classify the measures on sl k r sl k z which are invariant and ergodic under the action of the group a of positive diagonal matrices with positive entropy we apply this to prove that the set of exceptions to littlewood s conjecture has hausdorff dimension zero

a ground state solution for an asymptotically periodic quasilinear schrodinger equation abstract this article is concerned with the existence of positive ground state solutions for an asymptotically periodic quasilinear schrodinger equation by using a change of variables the quasilinear problem is transformed into a semilinear one then we use a nehari type constraint to get the existence result

on a problem in simultaneous diophantine approximation schmidt s conjecture for any i j 0 with i j 1 let bad i j denote the set of points x y 2 r 2 for which maxfkqxk 1 i kqyk 1 j g c q for all q 2 n here c c x y is a positive constant our main result implies that any nite intersection of such sets has full dimension this settles a conjecture of wolfgang m schmidt in the theory of simultaneous diophantine approximation

the work of einsiedler katok and lindenstrauss on the littlewood conjecture this document is intended as a slightly expanded writeup of my anticipated talk at the ams current events bulletin in new orleans january 2007 it is a brief report on the work of einsiedler katok and lindenstrauss on the littlewood conjecture 5 it is not intended in any sense for specialists and is indeed aimed at readers without any specific background either in measure theory dynamics or number theory any reader with any background in ergodic theory will be better served by consulting either the original paper or one of the surveys written by those authors see 7 and 12

a spectral method for the numerical simulation of quantum tunneling phenomena tunneling effects in solid state physics are frequently described via a quantum mechanical formulation based on the wigner distribution function in this paper a mixed finite difference spectral collocation method for the solution of the corresponding equation of motion is developed and analyzed numerical examples illustrating the convergence results are presented

operator splitting methods applied to spectral discretizations of quantum transport equations the wigner poisson equation is the most successful basis for transient simulations of quantum effect semiconductor devices so far we present a full discretization of this nonlinear pseudo differential system using a mixed spectral collocation and operator splitting method convergence and nonlinear stability of the scheme are proven

existence of solutions for quasilinear elliptic equations with hardy potential in this paper we consider the following quasilinear elliptic equation with hardy potential and dirichlet boundary condition i j 1ndj aij x u diu 12 i j 1ndsai j x u diudju x 2u f x u in where n n 3 is a smooth bounded domain di xi dsaij x s saij x s and 0 n 22 2 and x 2 is called the hardy potential by using the perturbation method we prove the existence of infinitely many solutions for the above problem

existence of nontrivial solutions for quasilinear elliptic equations at critical growth abstract in this paper we study the existence of nontrivial weak solutions for the following quasilinear elliptic equation with natural growth d j a ij x u d i u 1 2 s a ij x u d i ud j u f x u in a bounded smooth domain r n where aij x u are functions possibly unbounded with respect to u

the distribution of periodic torus orbits on homogeneous spaces we prove results towards the equidistribution of certain families of periodic torus orbits on homogeneous spaces with particular focus on the case of the diagonal torus acting on quotients of pgln r after attaching to each periodic orbit an integral invariant the discriminant our results have the following flavour certain standard conjectures about the distribution of such orbits hold up to exceptional sets of at most o orbits of discriminant the proof relies on the well separatedness of periodic orbits together with measure rigidity for torus actions we also give examples of sequences of periodic orbits of this action that fail to become equidistributed even in higher rank we give an application of our results to sharpen a theorem of minkowski on ideal classes in totally real number fields of cubic and higher degrees

numerical simulation of the quantum liouville poisson system abstract we simulate the one dimensional quantum liouville poisson system using the splitting scheme and the accompanying double fourier transformation in x and p space this code is used to study the quantum effects in the one dimensional electrostatic plasma i e the well known nonlinear landau damping and two stream instability problems