blow up of critical semilinear wave equations on schwarzschild spacetime abstract in this paper we established the blow up theorem for critical semilinear wave equations with focusing nonlinear term on schwarzschild spacetime concavity method is used to prove the main result which was introduced by levine payne in the papers levine and payne 1974 11 12 and levine 1973 7 in 1970s also a new auxiliary function with parameter is constructed following the idea from payne 1975 13 in order to guarantee that the result holds without any assumption on the initial data and initial energy

nonlinear scattering theory at low energy abstract we study the scattering theory of a conservative nonlinear one parameter group of operators on a hilbert space x relative to a group of linear unitary operators under certain hypotheses the scattering operator carries a neighborhood of 0 in x into x the theory is designed to apply to the semilinear schrodinger and klein gordon equations

nonexistence of global solutions to critical semilinear wave equations in exterior domain in high dimensions abstract the aim of this paper is to study the long time behavior of solutions to the initial boundary value problem of critical semilinear wave equations with small data in exterior domain in high dimensions n 5 we prove that solutions cannot exist globally in time the novelty is that we first use a cutoff function to transform the exterior problem to a cauchy problem then we divide the time into two intervals in one time interval we establish an energy estimate to control the error terms while in the other interval the asymptotic behavior of two test functions is used furthermore we obtain the upper bound of the lifespan by following the idea of zhou and han 2014

blow up of solutions to semilinear wave equations with variable coefficients and boundary this paper is devoted to studying initial boundary value problems for semilinear wave equations and derivative semilinear wave equations with variable coefficients on exterior domain with subcritical exponents in n space dimensions we will establish blow up results for the initial boundary value problems it is proved that there can be no global solutions no matter how small the initial data are and also we give the life span estimate of solutions for the problems

zeroes of the spectral density of the schroedinger operator with the slowly decaying wigner von neumann potential we consider the schrodinger operator mathcal l alpha on the half line with a periodic background potential and a perturbation which consists of two parts a summable potential and the slowly decaying wigner von neumann potential frac c sin 2 omega x delta x gamma where gamma in frac 1 2 1 the continuous spectrum of this operator has the same band gap structure as the continuous spectrum of the unperturbed periodic operator in every band there exist two points called critical where the eigenfunction equation has square summable solutions every critical point nu cr is an eigenvalue of the operator mathcal l alpha for some value of the boundary parameter alpha alpha cr specific to that particular point we prove that for alpha ne alpha cr the spectral density of the operator mathcal l alpha has a zero of the exponential type at nu cr

lifespan estimates of solutions to quasilinear wave equations with scattering damping abstract the formation of singularity of solutions to the quasilinear wave equations with scattering damping and divergence form nonlinearities is investigated the upper bound lifespan estimates of solutions to the problems with variable coefficient in the critical case and constant coefficient in the sub critical case are established by applying the rescaled test function method and iterative method

weighted strichartz estimates and global existence for semilinear wave equations in this paper we prove a sharp global existence theorem in all dimensions for nonlinear wave equations with power type nonlinearities the proof is based on a weighted strichartz estimate involving powers of the lorentz distance

spectral concentration and rapidly decaying potentials abstract we consider the spectral function 0 for the sturm liouville equation y q y 0 on 0 with a boundary condition y 0 cos y 0 sin 0 and where q l 0 a point 0 is called a point of spectral concentration if has a local maximum there we give a new formula for in terms of the prufer transformation angle and we identify a rapid transitional property of in the neighbourhood of points of spectral concentration this property provides a sensitive test of the existence of such points and lends itself to a detailed computational investigation which reveals such points in greater profusion than shown by for example sledge our approach also leads to the possibility of a theoretical investigation

an elementary proof of the blow up for semilinear wave equation in high space dimensions abstract this paper concerns the blow up of solutions to u tt u u p in high dimensions for n 4 and 1 p p 0 n where p 0 n is a critical exponent we proved that the solutions blow up in finite time by estimating the solutions near the wave front using elementary inequalities

finite time blow up for critical wave equations in high dimensions we prove that solutions to the critical wave equation 1 1 with dimension n 4 can not be global if the initial values are positive somewhere and nonnegative this completes the solution to the famous strauss conjecture about semilinear wave equations of the form u t2u u p 0 the rest of the cases the lower dimensional case n 3 and the sub or super critical cases were settled many years earlier by the work of several authors

asymptotic behavior of jost functions near resonance points for wigner von neumann type potentials in this work are considered radial schrodinger operators v r e where v r a sin br r w r with w r bounded w r o r 2 at infinity a b real the asymptotic behavior of the jost function and the scattering matrix near the resonance point e0 b2 4 are studied if a b then this point may be an eigenvalue embedded in the continuous spectrum the leading behavior of the jost function for all values of a and b were determined somewhat surprisingly situations were found where the jost function becomes singular as e e0 even if e0 is an embedded eigenvalue moreover it was found that the scattering matrix is always discontinuous at e0 except in a few special cases it is also shown that the asymptotics for the jost function and the scattering matrix hold under weaker assumptions on w r in particular an angular momentum term l l 1 r 2 may be incorporated into w r the results were also applied to a whole line problem with a potential v x such that v x 0 for x 0 and v x of wigner von neumann

weyl titchmarsh type formula for periodic schrodinger operator with wigner von neumann potential the schrodinger operator on the half line with periodic background potential perturbed by a certain potential of wigner von neumann type is considered the asymptotics of generalized eigenvectors for lambda is an element of c and on the absolutely continuous spectrum is established the weyl titchmarsh type formula for this operator is proven

jost functions and jost solutions for jacobi matrices i a necessary and sufficient condition for szegő asymptotics we provide necessary and sufficient conditions for a jacobi matrix to produce orthogonal polynomials with szeg asymptotics off the real axis a key idea is to prove the equivalence of szeg asymptotics and of jost asymptotics for the weyl solution we also prove l2 convergence of szeg asymptotics on the spectrum

supertransparent potentials for the dirac equation an extension of the classical darboux transformations is applied to the one dimensional dirac equation in order to construct von neumann wigner potentials allowing embedded eigenvalues these potentials lead to a novel type of scattering problem with a trivial s matrix composed of vanishing reflection coefficients and a trivial transmission coefficient related topics like the underlying symmetry of the dirac equation and the connection with positon solutions of nonlinear evolution equations are discussed

on the spectral theory of singular indefinite sturm liouville operators we consider a singular sturm liouville differential expression with an indefinite weight function and we show that the corresponding self adjoint differential operator in a krein space locally has the same spectral properties as a definitizable operator

the equation utt δu u p for the critical value of p the equation u tt u u p is considered in two and three space dimensions smooth cauchy data of compact support are given at t 0 for the case of three space dimensions john has shown that solutions with sufficiently small data exist globally in time if but that small data solutions blow up in finite time if glassey has shown the two dimensional case is similar this paper shows that small data solutions blow up in finite time when p is the critical value in three dimensions and in two

stokes phenomenon and the absolutely continuous spectrum of one dimensional schrodinger operators it is well known that the airy functions ai x and bi x form a fundamental set of solutions for the differential equation lu x u x xu x u x 0 x r and that the spectrum of the associated selfadjoint operator consists of the whole real axis and is purely absolutely continuous for any choice of boundary condition at x 0 also widely known is the fact that the semi axis is an anti stokes line for solutions of the differential equation lu z u z z c for each fixed value of the spectral parameter in this paper we show that this connection between the existence of anti stokes lines on the real axis and points of the absolutely continuous spectrum holds under much more general circumstances further correlations relating the stokes phenomenon to subordinacy properties of solutions of lu u at infinity and to the boundary behaviour of the titchmarsh weyl m function on the real axis are also deduced

negaton and positon solutions of the kdv and mkdv hierarchy we give a systematic classification and a detailed discussion of the structure motion and scattering of the recently discovered negaton and positon solutions of the korteweg de vries hierarchy there are two distinct types of negaton solutions which we label and where n 1 is the order of the wronskian used in the derivation for negatons the number of singularities and zeros is finite and they show very interesting time dependence the general motion is in the positive x direction except for certain negatons which exhibit one oscillation around the origin in contrast there is just one type of positon solution which we label for positons one gets a finite number of singularities for n odd but an infinite number for even values of n the general motion of positons is in the negative x direction with periodic oscillations negatons and positons retain their identities in a scattering process and their phase shifts are discussed we obtain a simple explanation of all phase shifts by generalizing the notions of mass and centre of mass to singular solutions finally it is shown that negaton and positon solutions of the kdv hierarchy can be used to obtain corresponding new solutions of the modified kdv hierarchy

weyl titchmarsh m function asymptotics local uniqueness results trace formulas and borg type theorems for dirac operators we explicitly determine the high energy asymptotics for weyl titchmarsh matrices associated with general dirac type operators on half lines and on r we also prove new local uniqueness results for dirac type operators in terms of exponentially small differences of weyl titchmarsh matrices as concrete applications of the asymptotic high energy expansion we derive a trace formula for dirac operators and use it to prove a borg type theorem

finite gap potentials and wkb asymptotics for one dimensional schrodinger operators consider the schrodinger operator h d2 dx2 v x with power decaying potential v x o x we prove that a previously obtained dimensional bound on exceptional sets of the wkb method is sharp in its whole range of validity the construction relies on pointwise bounds on finite gap potentials these bounds are obtained by an analysis of the jacobi inversion problem on hyperelliptic riemann surfaces

singular and maximal radon transforms analysis and geometry part 2 geometric theory 8 curvature introduction 8 1 three notions of curvature 8 2 theorems 8 3 examples 9 curvature some details 9 1 the exponential representation 9 2 diffeomorphism invariance 9 3 curvature condition cy 9 4 two lemmas 9 5 double fibration formulation 10 equivalence of curvature conditions 10 1 invariant submanifolds and deficient lie algebras 10 2 vanishing jacobians 10 3 construction of invariant submanifolds

on abstract strichartz estimates and the strauss conjecture for nontrapping obstacles we establish the strauss conjecture concerning small data global existence for nonlinear wave equations in the setting of exterior domains to compact obstacles for space dimensions n 3 and 4 the obstacle is assumed to be nontrapping and the solution is assumed to satisfy either dirichlet or neumann conditions along the boundary of the obstacle the key step in the proof is establishing certain abstract strichartz estimates for the linear wave equation on exterior domains

levinson s theorem for non local interactions levinson s theorem for a schrodinger equation with both local and non local symmetric potentials is studied in terms of the sturm liouville theorem a new convention for the phase shifts is applied instead of the usual one it is proved that the usual levinson theorem holds for the case with both local potential and non local symmetric cutoff potential which is not necessarily separable the problems related to the positive energy bound states and the physically redundant solutions are also discussed

nonexistence of global solutions to semilinear wave equations in high dimensions on considere le probleme de cauchy pour l equation d onde semi lineaire 2 t 2 u x t u x t p p 1 u x o f x u t x o g x avec f g c o r n

small unions of affine subspaces and skeletons via baire category abstract our aim is to find the minimal hausdorff dimension of the union of scaled and or rotated copies of the k skeleton of a fixed polytope centered at the points of a given set for many of these problems we show that a typical arrangement in the sense of baire category gives minimal hausdorff dimension in particular this proves a conjecture of r thornton our results also show that nikodym sets are typical among all sets which contain for every x r n a punctured hyperplane h x through x with similar methods we also construct a borel subset of r n of lebesgue measure zero containing a hyperplane at every positive distance from every point

absence of high energy spectral concentration for dirac systems with divergent potentials it is known that one dimensional dirac systems with potentials q which tend to or at infinity such that 1 q is of bounded variation have a purely absolutely continuous spectrum covering the whole real line we show that for the system on a half line there are no local maxima of the spectral density points of spectral concentration above some value of the spectral parameter if q satisfies certain additional regularity conditions these conditions admit thrice differentiable potentials of power or exponential growth the eventual sign of the derivative of the spectral density depends on the boundary condition imposed at the regular end point

absolute continuity and spectral concentration for slowly decaying potentials abstract we consider the spectral function 0 for the sturm liouville equation y qq y 0 on 0 with the boundary condition y 0 0 and where q has slow decay o x a a 0 as x we develop our previous methods of locating spectral concentration for q with rapid exponential decay this journal 81 1997 333 348 to deal with the new theoretical and computational complexities which arise for slow decay

long time existence for small amplitude semilinear wave equations we prove that a certain class of semilinear wave equations has global solutions if the initial data is small these existence results rely on mixed norm angular radial space time estimates more specifically we consider power nonlinearities u f p u where f p u u p and 2 t denotes the d alembertian on inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 01i in 1979 john investigated this equation for small initial data in dimension n 3 he proved that if p 1 2 then there is a global solution whereas for p n dimensions and the critical power p c above which global solutions exist should be the positive root of n 1 p 2 c n 1 p c 2 0 sideris proved that for all n the solutions can blow up if p p c recently zhou proved that for n 4 one has global solutions if p p c 2 here we prove that for n 8 one has global solutions if p p c furthermore we proved that for all n one has global solutions if p p c and the initial data are spherically symmetric in the radial case we use l p estimates derived from the explicit form of the fundamental solution using inequalities from classical analysis in the nonradial case we write the fundamental solution in polar coordinates and use fourier integral and special function estimates to handle the angular part

hilbert integrals singular integrals and radon transforms i on introduit deux nouvelles classes d operateurs on etudie leur relation et on montre comment on peut les appliquer a l etude des problemes aux valeurs limites

finite time blow up to critical semilinear wave equation outside the ball in 3 d abstract in this paper we study the initial boundary value problem of critical semilinear wave equation with small initial data in 3 d blow up result will be established no matter how small the data are we find a radial and positive solution of the corresponding linear problem then our blow up result follows from the idea of zhou and han 2014 what is more we get the upper bound of the lifespan estimate

fine structure of the zeros of orthogonal polynomials iv a priori bounds and clock behavior we prove locally uniform spacing for the zeros of orthogonal polynomials on the real line under weak conditions jacobi parameters approach the free ones and are of bounded variation we prove that for ergodic discrete schrodinger operators poisson behavior implies a positive lyapunov exponent both results depend on a priori bounds on eigenvalue spacings for which we provide several proofs

exact behavior of jost functions at low energy for schrodinger operators with central potential q r and angular momentum l the behavior of the jost function fl k as k 0 is investigated it is assumed that 0dr 1 r q r where 1 situations where q is integrable with 1 2 but not with 2 are of particular interest for potentials satisfying q r q0r 2 e 0 e 1 and l 0 the leading behavior of f0 k and the phase shift 0 k as k 0 is derived also comments are made on the differentiability properties of the jost solutions with respect to the variable k at k 0 for 1 levinson s theorem is proved thereby clarifying some questions raised recently by newton j math phys 27 2720 1986

spectral concentration and perturbed discrete spectra abstract we examine spectral concentration for a class of sturm liouville problems on 0 a typical example being y x x cx2 y x 0 the discrete spectrum for c 0 leads to spectral concentration in the continuous spectrum for c 0 and we use a new formula for the spectral function to make a detailed computational investigation of the way in which spectral concentration occurs in particular we find that as c decreases spectral concentration arises first from the lowest unperturbed eigenvalue and then in turn from these eigenvalues in increasing order of size

efficient calculation of spectral density functions for specific classes of singular sturm liouville problems new families of approximations to sturm liouville spectral density functions are derived for cases where the potential function has one of several specific forms this particular form dictates the type of expansion functions used in the approximation error bounds for the residuals are established for each case in the case of power potentials the approximate solutions of an associated terminal value problem at are shown to be asymptotic power series expansions of the exact solution numerical algorithms have been implemented and several examples are given demonstrating the utility of the approach

on subordinacy and analysis of the spectrum of one dimensional schrodinger operators abstract by considering the behaviour as n of the ratio of l2 0 n norms of solutions of d 2 dr 2 v r u xu 0 r x r a characterisation of the absolutely continuous and singular spectra of one dimensional schrodinger operators is deduced the analysis is applicable to all operators for which l d 2 dr 2 v r is regular at 0 and in the limit point case at infinity with v r locally integrable

an inverse problem in coupled mode theory we study an inverse problem for the zakharov shabat system which is motivated by an application to the design of co directional couplers with prescribed response properties

the primes contain arbitrarily long arithmetic progressions we prove that there are arbitrarily long arithmetic progressions of primes there are three major ingredients the first is szemeredi s theorem which asserts that any subset of the integers of positive density contains progressions of arbitrary length the second which is the main new ingredient of this paper is a certain transference principle this allows us to deduce from szemeredi s theorem that any subset of a suficiently pseudorandom set or measure of positive relative density contains progressions of arbitrary length the third ingredient is a recent result of goldston and yildirim which we reproduce here using this one may place a large fraction of the primes inside a pseudorandom set of almost primes or more precisely a pseudorandom measure concentrated on almost primes with positive relative density

hilbert integrals singular integrals and radon transforms ii etude des transformations de radon singulieres et des integrales de hilbert appliquees aux problemes aux valeurs limites probleme de neumann

the m function for hamiltonians with wigner von neumann potentials hamiltonians with strong wigner von neumann potentials are known to possess bound states in the continuous spectrum for smaller couplings these bound states become resonances which appear as singularities of the m function these singularities are intimately connected with the decay of the corresponding subordinate eigenfunction for coulomb type potentials with schrodinger operators this has recently be shown by atkinson the author and hinton klaus and shaw here these results are extended to a general class of wigner von neumann potentials for dirac and schrodinger operators

maximal operators and differentiation theorems for sparse sets we study maximal averages associated with singular measures on r our main result is a construction of singular cantor type measures supported on sets of hausdorff dimension 1 e with 0 e 1 e 1 e as a consequence we are able to answer a question of aversa and preiss on density and differentiation theorems for singular measures in one dimension our proof combines probabilistic techniques with the methods developed in multidimensional euclidean harmonic analysis in particular there are strong similarities to bourgain s proof of the circular maximal theorem in two dimensions

asymptotic integration of linear differential equations abstract we establish a new result on the asymptotic integration of linear differential systems of the form y r y where is diagonal and r is small more specifically we allow for different scales of asymptotic behaviour of the individual matrix elements of and r we also discuss an application to a self adjoint fourth order equation

concerning the strauss conjecture for the subcritical and critical cases on the exterior domain in two space dimensions abstract the aim of this paper is to verify the strauss conjecture for semilinear wave equations with variable coefficients and for the subcritical and critical cases on the exterior domain for when the number of space dimensions n 2 the main ingredients include generalized sideris ode blow up results and the existence and a priori estimates of the test functions see section 2 for this purpose we study the following initial boundary value problem for two dimensional semilinear wave equations with variable coefficients on the exterior domain with subcritical and critical exponents 0 1 u t t i j 1 2 i a i j x j u u p x t c 0 n 2 where u u x t is a real valued scalar unknown function in c 0 where is a smooth compact obstacle in r 2 and c is its complement two dimensional initial boundary value problems for semilinear wave equations with variable coefficients on the exterior domain are difficult to deal with because the fundamental solution of the harmonic equation is different from that for the higher dimensional case and thus there will be a growth factor in the logarithmic term in the key test function 0 see section 2 one of the key points of our paper is the improvement of a well known ode blow up result lemma 2 1 of 12 p 386 for when the differential inequality involves a logarithmic term and we obtain an estimate of the upper bound for the blow up time on the basis of the key results lemma 2 3 lemma 2 4 obtained we will establish a blow up result for the above initial boundary value problem it is proved that there can be no global solutions no matter how small the initial data are and a lifespan estimate for the solutions for this problem will also be given

strichartz estimates in the hyperbolic space and global existence for the semilinear wave equation the aim of this article is twofold first we consider the wave equation in the hyperbolic space hi and obtain the counterparts of the strichartz type estimates in this context next we examine the relationship between semilinear hyperbolic equations in the minkowski space and in the hyperbolic space this leads to a simple proof of the recent result of georgiev lindblad and sogge on global existence for solutions to semilinear hyperbolic problems with small data shifting the space time strichartz estimates from the hyperbolic space to the minkowski space yields weighted strichartz estimates in rn x 1r which extend the ones of georgiev lindblad and sogge

maximal averages over surfaces letmf x supt 0 f t d x denote the maximal operator associated with surface measured on a smooth surfaces we prove that ifsis convex and has finite order contact with its tangent lines then m is bounded onlp rn p 2 if and only ifd x h 1 l1 loc s for all tangent planeshnot passing through the origin letm f x supt 0 f t d x be the maximal operator associated with a nonisotropic dilation tof surface measured we prove that m often behaves far better than m due to a rotational curvature in the time parametert

kirchhoff s rule for quantum wires we formulate and discuss one particle quantum scattering theory on an arbitrary finite graph with n open ends and where we define the hamiltonian to be minus the laplace operator with general boundary conditions at the vertices this results in a scattering theory with n channels the corresponding on shell s matrix formed by the reflection and transmission amplitudes for incoming plane waves of energy e 0 is given explicitly in terms of the boundary conditions and the lengths of the internal lines it is shown to be unitary which may be viewed as the quantum version of kirchhoff s law we exhibit covariance and symmetry properties it is symmetric if the boundary conditions are real also there is a duality transformation on the set of boundary conditions and the lengths of the internal lines such that the low energy behaviour of one theory gives the high energy behaviour of the transformed theory finally we provide a composition rule by which the on shell s matrix of a graph is factorizable in terms of the s matrices of its subgraphs all proofs use only known facts from the theory of self adjoint extensions standard linear algebra complex function theory and elementary arguments from the theory of hermitian symplectic forms

value distribution and spectral analysis of differential operators the theory of value distribution is applied through a study of boundary value distribution for the weyl titchmarsh m function to the spectral analysis of differential operators on the half line particular consequences are the theoretical underpinning of a numerical approach to spectral analysis of differential operators exhibiting absolutely continuous or singular spectra and new identities for the spectral measure in terms of solutions of the associated differential equation

bounded solutions and absolute continuity of sturm liouville operators this work is a contribution to the theory of the absolutely continuous spectrum of one dimensional schriidinger operators i e sturm liouville operators generated by differential expressions of the form ru u qu in proofs of absolute continuity for such operators and also for the closely related dirac systems always asymptotic properties of solutions of zu lu for real or complex values of 1 have played an important role see for example 18 19 8 4 9 lo in these works and a number of others particular classes of potentials q were treated using properties of solutions in the course of the proof of absolute continuity somewhat more recently the connection between asymptotic properties of solutions and absolute continuity was given a theory i e results of the following form were proved let the solutions of zu lu satisfy some specific asymptotic property for a varying in some subset of r no or almost no a priori assumptions on q then operators generated by z are absolutely continuous in this set a first result of this type was given by carmona 3 who proved that local uniform boundedness of solutions together with their derivatives in some open a set implies pure absolute continuity in this set here we will mostly rely on the concept of non or racy of solutions which was shown by gilbert and pearson 7 12 61 to be a powerful tool in the investigation of the continuous spectrum of sturm liouville operators in section 2 we give some consequences of their results whi ch are useful in proofs of absolute continuity these consequences range from non existence of imbedded eigenvalues and the existence of absolutely continuous parts to pure absolute continuity and the existence of transient 210 0022 247x 92 5 00

the limiting absorption principle for the discrete wigner von neumann operator abstract we apply weighted mourre commutator theory to prove the limiting absorption principle for the discrete schrodinger operator perturbed by the sum of a wigner von neumann and long range type potential in particular this implies a new result concerning the absolutely continuous spectrum for these operators even for the one dimensional operator we show that methods of classical mourre theory based on differential inequalities and on the generator of dilation cannot apply to the aforementioned schrodinger operators

a limiting absorption principle for schrodinger operators with oscillating potentials part ii δ c sin b x x v x for short range v and small coupling constant c

local spectral properties of reflectionless jacobi cmv and schrodinger operators we prove that jacobi cmv and schrodinger operators which are reflectionless on a homogeneous set e in the sense of carleson under the assumption of a blaschke type condition on their discrete spectra accumulating at e have purely absolutely continuous spectrum on e

a geometric proof of the circular maximal theorem is bounded on l r for p 2 simple examples show that this fails for p 2 in this note we derive bourgain s result by geometric and combinatorial methods in particular we do not use the fourier transform in any way our proof is based on a combinatorial argument from 6 which in turn uses marstrand s three circle lemma 7 and a lemma involving two circles that seems to originate in 10 the three circle lemma was used in 7 to prove the following result which is a simple consequence of bourgain s theorem suppose a planar set e has the property for every point in the plane e contains some circle with that point as center then e has positive measure thus we show here that marstrand s lemma in combination with other ideas does indeed allow one to establish the stronger maximal function estimate furthermore we demonstrate in section 4 how to obtain the entire known range of l l estimates for the circular maximal function which is optimal possibly up to endpoints see 9 and 11 by using the methods from sections 2 and 3 one perhaps significant distinction from the techniques developed in 1 8 11 and 12 which involve the fourier transform is the fact that the methods presented here do not seem to yield estimates for the global maximal function mf x sup 0 t

power law bounds on transfer matrices and quantum dynamics in one dimension ii abstract we establish quantum dynamical lower bounds for a number of discrete one dimensional schrodinger operators these dynamical bounds are derived from power law upper bounds on the norms of transfer matrices we develop further the approach from part i and study many examples particular focus is put on models with finitely or at most countably many exceptional energies for which one can prove power law bounds on transfer matrices the models discussed in this paper include substitution models sturmian models a hierarchical model the prime model and a class of moderately sparse potentials

a remark on von neumann wigner type potentials a general formulation of the modulation function approach to von neumann wigner type potentials is given covering recent discussions of such potentials as special cases

generalized prufer variables for perturbations of jacobi and cmv matrices abstract prufer variables are a standard tool in spectral theory developed originally for perturbations of the free schrodinger operator we adapt the generalized prufer variables introduced by kiselev remling simon for perturbations of an arbitrary schrodinger operator to the setting of jacobi and szeg recursions we present an application to random l 2 perturbations of jacobi and cmv matrices and an application to decaying oscillatory perturbations of periodic jacobi and cmv matrices

resonances and analytic continuation for exponentially decaying sturm liouville potentials abstract the sturm liouville equation on the half line 0 is considered with a potential which decays exponentially at infinity it is known that green s function has an analytic continuation into part of the unphysical sheet im here it is shown that analytic continuation into the whole of this sheet is possible for certain types of decay the methods are constructive and they also provide a new and rigorous way of locating resonances