linear quadratic mean field control the invariant subspace method this note studies linear quadratic lq mean field social optimization problems with non uniform agents and indefinite state weight and analyzes the resulting linear system of ordinary differential equations odes subject to a partial initial condition and a growth condition we apply a subspace decomposition method for existence analysis and computation this is accomplished by carefully choosing the unspecified initial condition in the ode system such that the modes exceeding the admissible growth rate are not activated we further extend this method to lq mean field games

dynamic fracture of a rectangular limited permeable crack in magneto electro elastic media under a time harmonic elastic p wave abstract the dynamic behavior of a rectangular limited permeable crack deeply embedded in transversely isotropic magneto electro elastic media was investigated under an incident harmonic stress wave using the generalized almansi s theorem and the schmidt method with the help of the fourier transform this problem was formulated into three pairs of dual integral equations with the jumps in displacement across the crack surfaces as the unknown variables by directly expanding the jumps in displacement across the crack surfaces into a series of jacobi polynomials the solution to the dual integral equations was derived finally the dynamic response of a rectangular crack under a harmonic wave was analyzed and the effects of the electric permittivity and the magnetic permeability of air inside the crack the geometric shape of the rectangular crack and the characteristics of the harmonic wave on the stress the electric displacement and the magnetic flux intensity factors in magneto electro elastic media were concluded

stability of asymptotics of christoffel darboux kernels we study the stability of convergence of the christoffel darboux kernel associated with a compactly supported measure to the sine kernel under perturbations of the jacobi coefficients of the measure we prove stability under variations of the boundary conditions and stability in a weak sense under l 1 and random l 2 diagonal perturbations we also show that convergence to the sine kernel at x implies that x 0

stability of spectral types for jacobi matrices under decaying random perturbations abstract we study stability of spectral types for semi infinite self adjoint tridiagonal matrices under random decaying perturbations we show that absolutely continuous spectrum associated with bounded eigenfunctions is stable under hilbert schmidt random perturbations we also obtain some results for singular spectral types

multi dimensional schrodinger operators with some negative spectrum abstract we obtain conditions on the negative spectra of schrodinger operators with potentials v and v which guarantee that the positive real line is covered by the absolutely continuous spectrum

absolutely continuous spectrum of dirac operators with square integrable potentials we show that the absolutely continuous part of the spectral function of the one dimensional dirac operator on a half line with a constant mass term and a real square integrable potential is strictly increasing throughout the essential spectrum 1 1 the proof is based on estimates for the transmission coefficient for the full line scattering problem with a truncated potential and a subsequent limiting procedure for the spectral function furthermore we show that the absolutely continuous spectrum persists when an angular momentum term is added thus also establishing the result for spherically symmetric dirac operators in higher dimensions

schrodinger operators with sparse potentials asymptotics of the fourier transform of the spectral measure we study the pointwise behavior of the fourier transform of the spectral measure for discrete one dimensional schrodinger operators with sparse potentials we find a resonance structure which admits a physical interpretation in terms of a simple quasiclassical model we also present an improved version of known results on the spectrum of such operators

nonlinear fourier transform towards the construction of nonlinear fourier modes we study a version of the nonlinear fourier transform associated with zs akns systems this version is suitable for the construction of nonlinear analogues of fourier modes and for the perturbation theoretic study of their superposition we provide an iterative scheme for computing the inverse of our transform the relevant formulae are expressed in terms of bell polynomials and functions related to them in order to prove the validity of our iterative scheme we show that our transform has the necessary analytic properties we show that up to order three of the perturbation parameter the nonlinear fourier mode is a complex sinusoid modulated by the second bernoulli polynomial we describe an application of the nonlinear superposition of two modes to a problem of transmission through a nonlinear medium

the role of terminal cost reward in finite horizon discrete time lq optimal control the optimal control problem for time invariant linear systems with quadratic cost is considered for arbitrary i e non necessarily positive semidefinite terminal cost matrices a classification of such matrices is proposed based on the maximum horizon for which there is a finite minimum cost for all initial states when such an horizon is infinite the classification is further refined based on the asymptotic behavior of the optimal control law a number of characterizations and other properties of the proposed classification are derived in the study of the asymptotic behavior a characterization is given of those matrices a such that the image of ats0 converges in the gap metric for any subspace s0

a note on finite horizon lq problems with indefinite cost this paper addresses the discrete time finite horizon linear quadratic lq optimal control problem in the case in which the quadratic forms in the performance index are not assumed to be positive semidefinite but only symmetric we provide a necessary and sufficient condition for the existence of an optimal control

absence of singular continuous spectrum for perturbed discrete schrodinger operators abstract we show that the spectral measure of discrete schrodinger operators h u n u n 1 u n 1 v n u n does not have singular continuous component if the potential v n o n 1

the absolutely continuous spectrum of one dimensional schrodinger operators with decaying potentials we investigate one dimensional schrodinger operators with asymptotically small potentials it will follow from our results that if with then is an essential support of the absolutely continuous part of the spectral measure we also prove that if then the spectrum is purely absolutely continuous on these results are optimal

solutions spectrum and dynamics for schrodinger operators on infinite domains 1 introduction and main results in this paper we investigate the relations between the rate of decay of solutions of schrodinger equations continuity properties of spectral measures of the corresponding operators and dynamical properties of the corresponding quantum systems the first main result of this paper shows that in great generality certain upper bounds on the rate of growth of l 2 norms of generalized eigenfunctions over expanding balls imply certain minimal singularity of the spectral measures consider an operator h v defined by the differential expression h v v x

stress intensity factors of a rectangular crack meeting a bimaterial interface using the hypersingular integral equation method based on body force method a planar crack meeting the interface in a three dimensional dissimilar materials is analyzed the singularity of the singular stress field around the crack front terminating at the interface is analyzed by the main part analytical method of hypersingular integral equations then the numerical method of the hypersingular integral equation for a rectangular crack subjected to normal load is proposed by the body force method which the crack opening dislocation is approximated by the product of basic density functions and polynomials numerical solutions of the stress intensity factors of some examples are given

controlling parameter of the stress intensity factors for a planar interfacial crack in three dimensional bimaterials numerical solutions of singular integral equations are discussed in the analysis of a planar rectangular interfacial crack in three dimensional bimaterials subjected to tension the problem is formulated as a system of singular integral equations on the basis of the body force method in the numerical analysis unknown body force densities are approximated by the products of the fundamental density functions and power series where the fundamental density functions are chosen to express singular behavior along the crack front of the interface crack exactly the calculation shows that the present method gives smooth variations of stress intensity factors along the crack front for various aspect ratios the present method gives rapidly converging numerical results and highly satisfied boundary conditions throughout the crack boundary the stress intensity factors are given with varying the material combination and aspect ratio of the crack it is found that the stress intensity factors ki and kii are determined by the bimaterial constant e alone independent of elastic modulus ratio and poisson s ratio

sum rules and spectral measures of schrodinger operators with l 2 potentials necessary and sufficient conditions are presented for a positive measure to be the spectral measure of a half line schrodinger operator with square integrable potential

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

stabilization control for ito stochastic system with indefinite state and control weight costs in standard linear quadratic lq control the first step in investigating infinite horizon optimal control is to derive the stabilisation condition with the optimal lq controller this paper focus

the extended symplectic pencil and the finite horizon lq problem with two sided boundary conditions this technical note introduces a new approach to the solution of a very general class of finite horizon optimal control problems for discrete time systems this approach provides a parametric expression for the optimal control sequences as well as the corresponding optimal state trajectories by exploiting a new decomposition of the so called extended symplectic pencil this decomposition provides an original strategy for a more direct solution of the problem with no need of the system theoretic hypotheses including regularity of the symplectic pencil that have always been assumed in the literature so far

the generalised discrete algebraic riccati equation in linear quadratic optimal control this paper investigates the properties of the solutions of the generalised discrete algebraic riccati equation arising from the classic infinite horizon linear quadratic lq control problem in particular a geometric analysis is used to study the relationship existing between the solutions of the generalised riccati equation and the output nulling subspaces of the underlying system and the corresponding reachability subspaces this analysis reveals the presence of a subspace that plays an important role in the solution of the related optimal control problem which is reflected in the generalised eigenstructure of the corresponding extended symplectic pencil in establishing the main results of this paper several ancillary problems on the discrete lyapunov equation and spectral factorisation are also addressed and solved

first kdv integrals and absolutely continuous spectrum for 1 d schrodinger operator we consider 1 d schrodinger operators on l 2 r with slowly decaying potentials under some conditions on the potential related to the first integrals of the kdv equation we prove that the a c spectrum of the operator coincides with the positive semiaxis and the singular spectrum is unstable examples show that for special classes of sparse potentials these results can not be improved

the discrete algebraic riccati equation and linear matrix inequality we study the discrete time algebraic riccati equation in particular we show that even in the most general cases there exists a one one correspondence between solutions of the algebraic riccati equation and deflating subspaces of a matrix pencil we also study the relationship between algebraic riccati equation and the discrete time linear matrix inequality we show that in general only a subset of the set of rank minimizing solutions of the linear matrix inequality correspond to the solutions of the associated algebraic riccati equation and study under what conditions these sets are equal in this process we also derive very weak assumptions under which a riccati equation has a solution

maximal functions associated to filtrations abstract let t be a bounded linear or sublinear operator from lp y to lq x a maximal operator t f x supj t f yj x is associated to any sequence of subsets yj of y under the hypotheses that q p and the sets yj are nested we prove that t is also bounded classical theorems of menshov and zygmund are obtained as corollaries multilinear generalizations of this theorem are also established these results are motivated by applications to the spectral analysis of schrodinger operators

application of hypersingular integral equation method to three dimensional crack in electromagnetothermoelastic multiphase composites a three dimensional crack problem in electromagnetothermoelastic multiphase composites emte mcs under extended loads is investigated in this paper using green s functions the extended general displacement solutions are obtained by the boundary element method this crack problem is reduced to solving a set of hypersingular integral equations coupled with boundary integral equations in which the unknown functions are the extended displacement discontinuities then the behavior of the extended displacement discontinuities around the crack front terminating at the interface is analyzed by the main part analysis method of hypersingular integral equations analytical solutions for the extended singular stresses the extended stress intensity factors sifs and the extended energy release rate near the crack front in emte mcs are provided also a numerical method of the hypersingular integral equations for a rectangular crack subjected to extended loads is put forward with the extended displacement discontinuities approximated by the product of basic density functions and polynomials in addition distributions of extended sifs varying with the shape of the crack are presented the results show that the present method accurately yields smooth variations of extended sifs along the crack front

sum rules for jacobi matrices and their applications to spectral theory we discuss the proof of and systematic application of case s sum rules for jacobi matrices of special interest is a linear combination of two of his sum rules which has strictly positive terms among our results are a complete classification of the spectral measures of all jacobi matrices j for which j j 0 is hilbert schmidt and a proof of nevai s conjecture that the szego condition holds if j j 0 is trace class

wkb and spectral analysis of one dimensional schrodinger operators with slowly varying potentials consider a schrodinger operator on l 2 of the line or of a half line with appropriate boundary conditions if the potential tends to zero and is a finite sum of terms each of which has a derivative of some order in l 1 l p for some exponent p 2 then an essential support of the the absolutely continuous spectrum equals almost every generalized eigenfunction is bounded and satisfies certain wkb type asymptotics at infinity if moreover these derivatives belong to l p with respect to a weight x with 0 then the hausdorff dimension of the singular component of the spectral measure is strictly less than one

analysis of 3d fluid driven crack propagation problem in co seismic slip under p and s waves by hybrid hypersingular integral method this work reports a new and accurate way of theoretical and numerical description of the extended 3d fluid electromagnetic and flow driven crack progression in co seismic slip under p and s waves first based on the viscous fluid flow reciprocal work theorem the hybrid hypersingular integral equation hie method proposed by the author was defined by combined with the coupled extended wave time domain hie and the extended diffused interface phase field method the general extended 3d fluid flow velocity wave solutions are obtained by the extended wave time domains green s function method the 3d extended dynamic fluid driven crack modeling under fully coupled electromagnetothermoelastic p and s wave and flow field was established then the problem is reduced to solving a set of extended hybrid hies coupled with nonlinear boundary domain integral equations in which the unknown functions are the general extended flow velocity discontinuity waves the behavior of the general extended singular stress indices around the crack front terminating is analyzed by hybrid time domain main part analysis the general extended singular pore stress waves spsws and the extended dynamic stress intensity factors dsifs on the fluid driven crack surface are obtained from closed form solutions in addition a numerical method for the problem is proposed in which the extended velocity discontinuity waves are approximated by the product of time domain density functions and polynomials the extended dsifs and general extended spsws are calculated and the results are presented toward demonstrating the applicability of the proposed method

modified prufer and efgp transforms and deterministic models with dense point spectrum we provide a new proof of the theorem of simon and zhu that in the region e for a e energies d 2 dx 2 cos x 0 1 has lyapunov behavior with a quasi classical formula for the lyapunov exponent we also prove lyapunov behavior for a e e 2 2 for the discrete model with v j 2 e j v n 0 if n 1 4 9 the arguments depend on a direct analysis of the equations for the norm of a solution

imbedded singular continuous spectrum for schrodinger operators 1 1 hv v x and some self adjoint boundary condition at zero the operator 1 1 describes a charged particle such as an electron in the electric field v x when v x is decaying quickly one expects the spectral and dynamical properties of hy to remain close to those of the free operator ho recall that with the operator 1 1 one can associate in a canonical way a spectral measure p see for example 6 2 which contains much information about the quantum system the classical result in this direction going back to the beginning of the century is that if v g l1 the spectral measure on the positive semi axis remains purely absolutely continuous in one dimension an absolutely continuous spectrum corresponds to the ballistic rate of propagation of the particle a natural question to ask is what are the critical rates of decay at which some changes in the spectral and dynamical properties of hy may happen as far as v x has any decay at all the essential spectrum coincides with 0 oo but the quality of the spectrum and dynamics may change in 1928 wigner and von neumann 27 showed that there exist potentials v x satisfying v x jsuch that hy has positive imbedded eigenvalue e 1 this is a purely quantum resonance phenomenon since v can be chosen arbitrarily small and the bound state originates from long range correlations in the potential rather than the usual confining effect naboko 17 and simon 25 provided constructions which show that much more drastic changes are possible if the potential decays arbitrarily slower than coulomb namely for any positive function h x which grows at infinity there exist potentials v such that v x yjgand hy has a dense set of positive eigenvalues due to some recent results 3 21 it is known that the absolutely continuous spectrum on the positive semi axis is preserved for v o c l x l 2 e and in fact for v g l2 8 more precisely for such potentials the absolutely continuous part of the spectral measure iac gives positive weight to any subset of 0 oo of positive lebesgue measure therefore the naboko

one dimensional schrodinger operators with random decaying potentials we investigate the spectrum of the following random schrodinger operators h omega frac d 2 dt 2 a t f x t omega wheref xt is a markovian potential studied by the russian school 8 we completely describe the transition of the spectrum from pure point type to absolutely continuous type as the decreasing order ofa t grows this is an extension to a continuous case of the result due to delyon simon souillard 6 who deal with the lattice case

asymptotic green functions a generalized semiclassical approach for scattering by multiple barrier potentials we show how quantum mechanical barrier reflection and transmission coefficients and can be obtained from asymptotic green functions we exemplify our results by calculating such coefficients for the rosen morse rm potential for multiple barrier potentials v x jv j x where each v j goes to zero for x we derive the asymptotic green functions by a generalized semiclassical approximation which is based on the usual sum over classical paths considered only in the classically allowed regions and includes local quantum effects through the individual j and j the approach is applied to double rm potentials and to woods saxon barriers we obtain analytical expressions for the transmission and reflection probabilities of these potentials which are very accurate when compared with exact numerical calculations being much better than the usual wkb approximation finally we briefly discuss how to extend the present method to other kinds of potential

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

absolutely continuous spectrum of schrodinger operators with potentials slowly decaying inside a cone abstract for a large class of multi dimensional schrodinger operators it is shown that the absolutely continuous spectrum is essentially supported by 0 we require slow decay and mildly oscillatory behavior of the potential in a cone and can allow for arbitrary non negative bounded potential outside the cone in particular we do not require the existence of wave operators the result and method of proof extends previous work by laptev naboko and safronov

wkb asymptotic behavior of almost all generalized eigenfunctions for one dimensional schrodinger operators with slowly decaying potentials we prove the wkb asymptotic behavior of solutions of the differential equation d2u dx2 v x u eu for a e e a where v v1 v2 v1 lp r and v2 is bounded from above with a lim supx v x while v 2 x lp r 1 p 2 these results imply that schrodinger operators with such potentials have absolutely continuous spectrum on a we also establish wkb asymptotic behavior of solutions for some energy dependent potentials

schrodinger operators in the twentieth century this paper reviews the past fifty years of work on spectral theory and related issues in nonrelativistic quantum mechanics

criteria for eigenvalues embedded into the absolutely continuous spectrum of perturbed stark type operators abstract in this paper we consider the perturbed stark operator h u h 0 u q u u x u q u where q is the power decaying perturbation the criteria for q such that h h 0 q has at most one eigenvalue finitely many infinitely many eigenvalues are obtained all the results are quantitative and are generalized to the perturbed stark type operator

modified prufer and efgp transforms and the spectral analysis of one dimensional schrodinger operators using control of the growth of the transfer matrices wediscuss the spectral analysis of continuum and discrete half line schrodinger operators with slowly decaying potentials among our results we show if where w has compact support and then h has purely a c resp purely s c spectrum on 0 if for n 1 2 a n potentials where a n are independent identically distributed random variables with e a n 0 e a 2 n 1 and 2 we find singular continuous spectrum with explicitly computable fractional hausdorff dimension

on the absolutely continuous spectrum of stark operators the stability of the absolutely continuous spectrum of the one dimensional stark operator under perturbations of the potential is discussed the focus is on proving this stability under minimal assumptions on smoothness of the perturbation a general criterion is presented together with some applications these include the case of periodic perturbations where we show that any perturbation v l 1 h 1 2 preserves the a c spectrum

one dimensional schrodinger operators with random or deterministic potentials new spectral types abstract we prove sufficient conditions involving only potential asymptotic near one of the infinities in order to have purely absolutely continuous components in the spectrum these deterministic results are then applied to random cases and exhibit classes of models for which with probability one one component of the spectrum is purely absolutely continuous and the rest is dense pure point with exponentially decaying eigenfunctions

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

absolutely continuous spectrum of one dimensional schrodinger operators and jacobi matrices with slowly decreasing potentials we prove that for any one dimensional schrodinger operator with potentialv x satisfying decay condition v x cx 3 4 e the absolutely continuous spectrum fills the whole positive semi axis the description of the set in on which the singular part of the spectral measure might be supported is also given analogous results hold for jacobi matrices

sparse 1d discrete dirac operators i quantum transport abstract some dynamical lower bounds for one dimensional discrete dirac operators with different classes of sparse potentials are presented and the particular role of the particle mass is emphasized

transient and recurrent spectrum abstract we deal primarily with spectral analysis of an abstract self adjoint operator h on a hilbert space h we propose a further refinement of the absolutely continuous subspace h ac into the transient absolutely continuous subspace h tac which is the closure of those with e ith o t n for all n and the recurrent absolutely continuous subspace h rac h ac h tac we discuss general features of this breakup in a subsequent paper we construct analytic almost periodic functions v on so that h d 2 dx 2 v x on l 2 has only recurrent absolutely continuous spectrum in the sense that h rac h

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

on the absolutely continuous spectrum of one dimensional schrodinger operators with square summable potentials for continuous and discrete one dimensional schrodinger operators with square summable potentials the absolutely continuous part of the spectrum is essentially supported by 0 and 2 2 respectively this fact is proved by considering a priori estimates for the transmission coefficient