spectral statistics for unitary transfer matrices of binary graphs quantum graphs have recently been introduced as model systems to study the spectral statistics of linear wave problems with chaotic classical limits it is proposed here to generalize this approach by considering arbitrary directed graphs with unitary transfer matrices an exponentially increasing contribution to the form factor is identified when performing a diagonal summation over periodic orbit degeneracy classes a special class of graphs so called binary graphs is studied in more detail for these the conditions for periodic orbit pairs to be correlated including correlations due to the unitarity of the transfer matrix can be given explicitly using combinatorial techniques it is possible to perform the summation over correlated periodic orbit pair contributions to the form factor for some low dimensional cases gradual convergence towards random matrix results is observed when increasing the number of vertices of the binary graphs

two point spectral correlations for star graphs the eigenvalues of the schrodinger operator on a graph g are related via an exact trace formula to periodic orbits on g this connection is used to calculate two point spectral statistics for a particular family of graphs called star graphs in the limit as the number of edges tends to infinity combinatorial techniques are used to evaluate both the diagonal same orbit and off diagonal different orbit contributions to the sum over pairs of orbits involved in this way a general formula is derived for terms in the short time expansion of the form factor k in powers of and the first few are computed explicitly the result demonstrates that k is neither poissonian nor random matrix but an intermediate between the two off diagonal pairs of orbits are shown to make a significant contribution to all but the first few coefficients

h 1 scattering for systems of n defocusing weakly coupled nls equations in low space dimensions abstract we prove that the scattering operators and wave operators are well defined in the energy space for the system of defocusing schrodinger equations i t u u 1 n u p 1 u p 1 u 0 1 n u 0 1 n u 0 1 n h 1 r d n with n 2 0 0 for p 2 if d 1 p 1 if d 2 and 1 p 2 if d 3

local smoothing properties of dispersive equations is it possible for time evolution partial differential equations which are reversible and conservative to smooth locally the initial data for the linear wave equation for instance the answer is no however in 10 t kato found a local smoothing property of the korteweg de vries equation the solution of the initial value problem is locally one derivative smoother than the initial datum kato s proof uses in a curcial way the algebraic properties of the symbol for the korteweg de vries equation and the fact that the underlying spatial dimension is one actually judging from the way several integrations by parts and cancellations conspire to reveal a smoothing effect one would be inclined to believe this was a special property of the k dv equation this is not however the case in this paper we attempt to describe a general local smoothing effect for dispersive equations and systems the smoothing effect is due to the dispersive nature of the linear part of the equation all the physically significant dispersive equations and systems known to us have linear parts displaying this local smoothing property to mention only a few the k dv benjamin ono intermediate long wave various boussinesq and schrodinger equations are included we study thus equations and systems of the form

optimal control of deterministic systems via shifted legendre polynomials the synthesis of an optimal control function for deterministic systems described by integrodifferential equations is investigated by using the elegant operational properties of shifted legendre polynomials a direct computational algorithm for evaluating the optimal control and trajectory of deterministic systems is developed an example is given to illustrate the utility of this method

regular quantum graphs we introduce the concept of regular quantum graphs and construct connected quantum graphs with discrete symmetries the method is based on a decomposition of the quantum propagator in terms of permutation matrices which control the way incoming and outgoing channels at vertex scattering processes are connected symmetry properties of the quantum graph as well as its spectral statistics depend on the particular choice of permutation matrices also called connectivity matrices and can now be controlled easily the method may find applications in the study of quantum random walks and may also prove to be useful in analysing universality in spectral statistics

non trapping magnetic fields and morrey campanato estimates for schrodinger operators abstract we prove some uniform in a priori estimates for solutions of the equation i a 2 u v x u i u f 0 0 the estimates are obtained in terms of morrey campanato norms and can be used to prove absence of zero resonances in a suitable sense for electromagnetic hamiltonians quantitative conditions on the size of the trapping component of the magnetic field and the non repulsive component of the electric field are given

the forward problem for the electromagnetic helmholtz equation with critical singularities abstract we study the forward problem of the magnetic schrodinger operator with potentials that have a strong singularity at the origin we obtain new resolvent estimates and give some applications on the spectral measure and on the solutions of the associated evolution problem

generalized strichartz inequalities for the wave equation strichartz inequalities for the wave equation u f are estimates of the solution u of the cauchy problem for that equation in the form of space time integral norms in terms of similar norms of the inhomogeneity f and of suitable norms of the initial data

remarks on endpoint strichartz estimates for schrodinger equations with the critical inverse square potential abstract the purpose of this paper is to study the validity of global in time strichartz estimates for the schrodinger equation on r n n 3 with the negative inverse square potential x 2 in the critical case n 2 2 4 it turns out that the situation is different from the subcritical case n 2 2 4 in which the full range of strichartz estimates is known to be hold more precisely splitting the solution into the radial and non radial parts we show that i the radial part satisfies a weak type endpoint estimate which can be regarded as an extension to higher dimensions of the endpoint strichartz estimate with radial data for the two dimensional free schrodinger equation ii other endpoint estimates in lorentz spaces for the radial part fail in general iii the non radial part satisfies the full range of strichartz estimates

star graphs and seba billiards we derive an exact expression for the two point correlation function for quantum star graphs in the limit as the number of bonds tends to infinity this turns out to be identical to the corresponding result for certain eba billiards in the semiclassical limit the reasons for this are discussed the formula we derive is also shown to be equivalent to a series expansion for the form factor the fourier transform of the two point correlation function previously calculated using periodic orbit theory

scattering theory for nonlinear schrodinger equations with inverse square potential abstract we study the long time behavior of solutions to nonlinear schrodinger equations with some critical rough potential of a x 2 type the new ingredients are the interaction morawetz type inequalities and sobolev norm property associated with p a a x 2 we use such properties to obtain the scattering theory for the defocusing energy subcritical nonlinear schrodinger equation with inverse square potential in energy space h 1 r n

on the wave equation with a large rough potential we prove an optimal dispersive l decay estimate for a three dimensional wave equation perturbed with a large nonsmooth potential belonging to a particular kato class the proof is based on a spectral representation of the solution and suitable resolvent estimates for the perturbed operator

analysis and parameter identification of bilinear systems via shifted legendre polynomials the problems of analysis and parameter identification of lime delay systems have been recently studied using rectangular functions such as walsh and block pulse functions in this paper the continuous shifted legendre polynomials are first used to solve the above mentioned problems the approach followed is that of converting the delay differential equation to an algebraic form through using the operational matrices of integration and delay the key point is the derivation of a delay operational matrix d which relates the coefficient vector x of the shifted legendre series of a given function x t with the coefficient vector x of its delay form x t in the form x dx where the elements of a are obtained from the initial function of x t two computational examples are given to illustrate the utility of the method the availability of recursive algorithms for constructing the delay operational matrix d makes the method particularly attractive to practise

on the level spacing distribution in quantum graphs we derive a formula for the level spacing probability distribution in quantum graphs we apply it to simple examples and we discuss its relation with previous work and its possible application in more general cases moreover we derive an exact and explicit formula for the level spacing distribution of integrable quantum graphs

unitary stochastic matrix ensembles and spectral statistics we propose to study unitary matrix ensembles defined in terms of unitary stochastic transition matrices associated with markov processes on graphs we argue that the spectral statistics of such an ensemble after ensemble averaging depends crucially on the spectral gap between the leading and subleading eigenvalue of the underlying transition matrix it is conjectured that unitary stochastic ensembles follow one of the three standard ensembles of random matrix theory in the limit of infinite matrix size n if the spectral gap of the corresponding transition matrices closes slower than 1 n the hypothesis is tested by considering several model systems ranging from binary graphs to uniformly and non uniformly connected star graphs and diffusive networks in arbitrary dimensions

endpoint strichartz estimates we prove an abstract strichartz estimate which implies previously unknown endpoint strichartz estimates for the wave equation in dimension n 4 and the schrodinger equation in dimension n 3 three other applications are discussed local existence for a nonlinear wave equation and strichartz type estimates for more general dispersive equations and for the kinetic transport equation

l p l q estimates for electromagnetic helmholtz equation abstract in space dimension n 3 we consider the electromagnetic schrodinger hamiltonian h i a x 2 v and the corresponding helmholtz equation i a x 2 u u v x u f in r n we extend the well known l p l q estimates for the solution of the free helmholtz equation to the case when the electromagnetic hamiltonian h is considered

on the radius of spatial analyticity for semilinear symmetric hyperbolic systems abstract we study the problem of propagation of analytic regularity for semi linear symmetric hyperbolic systems we adopt a global perspective and we prove that if the initial datum extends to a holomorphic function in a strip of radius width e 0 the same happens for the solution u t for a certain radius e t as long as the solution exists our focus is on precise lower bounds on the spatial radius of analyticity e t as t grows we also get similar results for the schrodinger equation with a real analytic electromagnetic potential

finite time balanced truncation for linear systems via shifted legendre polynomials abstract in this paper we present a finite time model order reduction method for linear systems via shifted legendre polynomials the main idea of the approach is to use finite time empirical gramians which are constructed from impulse responses by solving block tridiagonal linear systems to generate approximate balanced system for the large scale system the balancing transformation is directly computed from the expansion coefficients of impulse responses in the space spanned by shifted legendre polynomials without individual reduction of the gramians and a separate eigenvector solve then the reduced order model is constructed by truncating the states corresponding to the small approximate hankel singular values hsvs the stability preservation of the reduced model is briefly discussed and in combination with the dominant subspace projection method we modify the reduction procedure to alleviate the shortcomings of the above method which may unexpectedly lead to unstable systems even though the original one is stable furthermore the properties of the resulting reduced models are considered finally numerical experiments are given to demonstrate the effectiveness of the proposed methods

lp lq estimates for electromagnetic helmholtz equation singular potentials abstract in space dimension n 3 we consider the electromagnetic schrodinger hamiltonian h i a x 2 v and the corresponding helmholtz equation i a x 2 u 1 i u v x u f in r n where the magnetic and electric potentials are allowed to have singularities at the origin and decay at infinity we extend the well known l p l q estimates for the solution of the free helmholtz equation to the case when the electromagnetic hamiltonian h is considered this work extends the results that appear in 7

uniform resolvent estimates for schrodinger operator with an inverse square potential abstract we study the uniform resolvent estimates for schrodinger operator with a hardy type singular potential let l v v x where is the usual laplacian on r n and v x v 0 r 2 where r x x x and v 0 c 1 s n 1 is a real function such that the operator v 0 n 2 2 4 is a strictly positive operator on l 2 s n 1 we prove some new uniform weighted resolvent estimates and also obtain some uniform sobolev estimates associated with the operator l v

endpoint strichartz estimates for the magnetic schrodinger equation abstract we prove strichartz estimates for the schrodinger equation with an electromagnetic potential in dimension n 3 the decay and regularity assumptions on the potentials are almost critical i e close to the coulomb case in addition we require repulsivity and a nontrapping condition which are expressed as smallness of suitable components of the potentials while the potentials themselves can be large the proof is based on smoothing estimates and new sobolev embeddings for spaces associated to magnetic potentials

time decay of scaling critical electromagnetic schrodinger flows we obtain a representation formula for solutions to schrodinger equations with a class of homogeneous scaling critical electromagnetic potentials as a consequence we prove the sharp l 1 to l infty time decay estimate for the 3d inverse square and the 2d aharonov bohm potentials

on the scattering problem for the nonlinear schrodinger equation with a potential in 2d abstract we consider the scattering problem for the nonlinear schrodinger equation with a potential in two space dimensions appropriate resolvent estimates are proved and applied to estimate the operator a s appearing in commutator relations the equivalence between the operators v s 2 and s 2 in l 2 norm sense for 0 s 1 is investigated by using free resolvent estimates and gaussian estimates for the heat kernel of the schrodinger operator v our main result guarantees the global existence of solutions and time decay of the solutions assuming initial data have small weighted sobolev norms moreover the global solutions obtained in the main result scatter

strichartz estimates and local smoothing estimates for asymptotically flat schrodinger equations abstract in this article we study global in time strichartz estimates for the schrodinger evolution corresponding to long range perturbations of the euclidean laplacian this is a natural continuation of a recent article d tataru parametrices and dispersive estimates for schrodinger operators with variable coefficients amer j math 130 2008 571 634 of the third author where it is proved that local smoothing estimates imply strichartz estimates by d tataru parametrices and dispersive estimates for schrodinger operators with variable coefficients amer j math 130 2008 571 634 the local smoothing estimates are known to hold for small perturbations of the laplacian here we consider the case of large perturbations in three increasingly favorable scenarios i without non trapping assumptions we prove estimates outside a compact set modulo a lower order spatially localized error term ii with non trapping assumptions we prove global estimates modulo a lower order spatially localized error term and iii for time independent operators with no resonance or eigenvalue at the bottom of the spectrum we prove global estimates for the projection onto the continuous spectrum

regular articlegeneralized strichartz inequalities for the wave equation we make a synthetic exposition of the generalized strichartz inequalities for the wave equation obtained in 6 together with the limiting cases recently obtained in 13 with as simple proofs as possible the proofs combine stationary phase estimates dyadic decompositions the hardy littlewood inequality or the holder inequality in time and abstract duality and interpolation arguments

the shifted legend re approach to non linear system analysis and identification abstract applications of the shifted legendre polynomial expansion to the analysis and identification of the non linear system described by a hammerstein model which consists of a single valued non linearity followed by a linear plant are studied for the analysis by using the shifted legendre polynomial expansion the solution of a non linear state equation is reduced to the solution of a linear algebraic matrix equation for the identification through the shifted legendre expansions of the measured input output data the unknown parameters of both the linear plant and the characterization of the non linear element are estimated using the least squares method algorithms are presented numerical examples are given to illustrate the use of this approach

improved time decay for a class of scaling critical electromagnetic schrodinger flows we consider a schrodinger hamiltonian h a a with scaling critical and time independent external electromagnetic potential and assume that the angular operator l associated to h is positive definite we prove the following if e ith a a l1 l t n 2 then x g n e ith a a x g n l1 l t n 2 g n g n being a positive number explicitly depending on the ground level of l and the space dimension n we prove similar results also for the heat semi group generated by h a a

magnetic virial identities weak dispersion and strichartz inequalities we show a family of virial type identities for the schrodinger and wave equations with electromagnetic potentials as a consequence some weak dispersive inequalities in space dimension n 3 involving morawetz and smoothing estimates are proved finally we apply them to prove strichartz inequalities for the wave equation with a non trapping electromagnetic potential with almost coulomb decay

spectral gap of doubly stochastic matrices generated from equidistributed unitary matrices to a unitary matrix u we associate a doubly stochastic matrix m by taking the squared modulus of each element of u to study the connection between onset of quantum chaos on graphs and ergodicity of the underlying markov chain specified by m we study the limiting distribution of the spectral gap of m when u is taken from the circular unitary ensemble and the dimension n of u is taken to infinity we prove that the limiting distribution is degenerate the gap tends to its maximal value 1 the shape of the gap distribution for finite n is also discussed

on the series expansion approach to the identification of hammerstein systems a polynomial identification algorithm for recovering a nonlinearity in the hammerstein system is proposed the estimate employs the legendre orthogonal system with adaptively selected number of terms the global consistency along with rates of convergence are established no assumptions concerning continuity of the nonlinearity or its functional form are made a data driven method using the cross validation technique for selecting the number of terms in the estimate is presented

analysis and optimal control of time varying systems via chebyshev polynomials abstract the operational properties of the forward and backward integration and the product of chebyshev polynomials are derived these properties are applied to the analysis and optimal control of time varying linear systems by the approximation of time functions by truncated chebyshev series due to the unique property of the chebyshev polynomials excellent results are obtained as demonstrated by examples

regularity of solutions to the schrodinger equation soit f dans l espace de schwartz s r n et soit s t f x u x t rn exp ix exp it 2 f d x r n t r f est la transformee de fourier de f on considere u solution de l equation de schrodinger u i u t on pose s f x sup 0 t 1 s t f x x r n on introduit des espaces de sobolev h s en posant h s f s f hd s s r ou f hd s rn 1 2 s f 2 d 1 2 on etudie pour quelles valeurs de s on a b s f x 2 dx 1 2 c b f hd s f s r n pour toutes les boucles b de r n

some dispersive estimates for schrodinger equations with repulsive potentials we prove the local smoothing effect for schrodinger equations with repulsive potentials for n 3 the estimates are global in time and are proved using a variation of morawetz multipliers as a consequence we give sharp constants to measure the attractive part of the potential and its rate of decay which turns out to be different whether dimension 3 or higher are considered also a notion of zero resonance arises in a natural way our smoothing estimate allows us to use sobolev inequalities and treat nonradial perturbations

semiclassical theory of spectral rigidity the spectral rigidity l of a set of quantal energy levels is the mean square deviation of the spectral staircase from the straight line that best fits it over a range of l mean level spacings in the semiclassical limit 0 formulae are obtained giving l as a sum over classical periodic orbits when l l max where l max n 1 for a system of n freedoms l is shown to display the following universal behaviour as a result of properties of very long classical orbits if the system is classically integrable all periodic orbits filling tori l 1 5 l as in an uncorrelated poisson eigenvalue sequence if the system is classically chaotic all periodic orbits isolated and unstable and has no symmetry l in l 2 2 d if 1 l l max as in the gaussian unitary ensemble of random matrix theory if the system is chaotic and has time reversal symmetry l in l 2 e if 1 l l max as in the gaussian orthogonal ensemble when l l max l saturates non universally at a value determined by short classical orbits of order n 1 for integrable systems and in 1 for chaotic systems these results are obtained by using the periodic orbit expansion for the spectral density together with classical sum rules for the intensities of long orbits and a semiclassical sum rule restricting the manner in which their contributions interfere for two examples l is studied in detail the rectangular billiard integrable and the riemann zeta function assuming its zeros to be the eigenvalues of an unknown quantum system whose unknown classical limit is chaotic

analysis of linear time varying systems and bilinear systems via fourier series a method for finding an approximate solution of linear time varying systems and bilinear systems via fourier series is proposed properties of fourier series are first briefly presented and the operational matrix for integration together with the product operational matrix are utilized to reduce the dynamical equation of a linear time varying system or bilinear system to a set of simultaneous linear algebraic equations illustrative examples are also given

dispersive estimates for schrodinger operators in dimensions one and three we consider l 1 l estimates for the time evolution of hamiltonians h v in dimensions d 1 and d 3 with bound t 2 we require decay of the potentials but no regularity in d 1 the decay assumption is 1 x v x dx whereas in d 3 it is v x c 1 x 3

scattering in the energy space for the nls with variable coefficients we consider the nls with variable coefficients in dimension n ge 3 begin aligned i partial t u lu f u 0 qquad lv nabla b cdot a x nabla b v c x v qquad nabla b nabla ib x end aligned on mathbb r n or more generally on an exterior domain with dirichlet boundary conditions for a gauge invariant defocusing nonlinearity of power type f u simeq u gamma 1 u we assume that l is a small long range perturbation of delta plus a potential with a large positive part the first main result of the paper is a bilinear smoothing interaction morawetz estimate for the solution as an application under the conditional assumption that strichartz estimates are valid for the linear flow e itl we prove global well posedness in the energy space for subcritical powers gamma 1 frac 4 n when the domain is mathbb r n by extending the strichartz estimates due to tataru am j math 130 3 571 634 2008 we prove that the conditional assumption is satisfied and deduce well posedness and scattering in the energy space

periodic orbit action correlations in the baker map periodic orbit action correlations are studied for the piecewise linear area preserving baker map semiclassical periodic orbit formulae together with universal spectral statistics in the corresponding quantum baker map suggest the existence of universal periodic orbit correlations the calculation of periodic orbit sums for the baker map can be performed with the help of a perron frobenius type operator this makes it possible to study periodic orbit correlations for orbits with period up to 500 iterations of the map periodic orbit correlations are found to agree quantitatively with the predictions from random matrix theory up to a critical length determined by the semiclassical error exponentially increasing terms dominate the correlations for longer orbits which are due to the violation of unitarity in the semiclassical approximation

legendre series approach to identification and analysis of linear systems in this paper the legendre operational matrix of integration is introduced and it is subsequently used for the identification and analysis of linear systems the algorithms proposed are easily implementable in a digital computer and they are analogous to those already derived for other types of orthogonal functions

time decay for solutions of schrodinger equations with rough and time dependent potentials in this paper we establish dispersive estimates for solutions to the linear schrodinger equation in three dimensions frac 1 i partial t psi triangle psi v psi 0 qquad psi s f 0 1 where v t x is a time dependent potential that satisfies the conditions sup t v t cdot l frac 3 2 mathbb r 3 sup x in mathbb r 3 int mathbb r 3 int infty infty frac v hat tau x x y d tau dy c 0 here c0 is some small constant and v hat tau x denotes the fourier transform with respect to the first variable we show that under these conditions 0 1 admits solutions l t l2x 3 l2t l6x 3 for any f l2 3 satisfying the dispersive inequality psi t infty le c t s frac32 f 1 text for all times t s 0 2 for the case of time independent potentials v x 0 2 remains true if int mathbb r 6 frac v x v y x y 2 dxdy 4 pi 2 text and v mathcal k sup x in mathbb r 3 int mathbb r 3 frac v y x y dy 4 pi we also establish the dispersive estimate with an e loss for large energies provided v mathcal k v 2 infty

dispersive estimates for the three dimensional schrodinger equation with rough potentials the three dimensional schrodinger propogator e ith h v is a bounded map from l 1 to l with norm controlled by t 3 2 provided the potential satisfies two conditions an integrability condition limiting the singularities and decay of v and a zero energy spectral condition on h this is shown by expressing the spectral measure of h in terms of its resolvents and proving a family of l p mapping estimates for the resolvents previous results in this direction had required v to satisfy explicit pointwise bounds

smoothing estimates for the variable coefficients schrodinger equation with electromagnetic potentials abstract in this paper we develop the classical multiplier technique to build up a virial identity for the electromagnetic variable coefficients schrodinger equation following the strategy of fanelli and vega 2009 13 we shall use such an identity to prove smoothing estimates for the associated flow in a perturbative setting