stability of the absolutely continuous spectrum of random schrodinger operators on tree graphs the subject of this work is random schrodinger operators on regular rooted tree graphs with stochastically homogeneous disorder the operators are of the form h t u v acting in l2 with t the adjacency matrix u a radially periodic potential and v a random potential this includes the only class of homogeneously random operators for which it was proven that the spectrum of h exhibits an absolutely continuous ac component a results established by a klein for weak disorder in case u 0 and v given by iid random variables on our main contribution is a new method for establishing the persistence of ac spectrum under weak disorder the method yields the continuity of the ac spectral density of h at 0 the latter is shown to converge in the l 1 sense over closed borel sets in which h 0 has no singular spectrum the analysis extends to random potentials whose values at different sites need not be independent assuming only that their joint distribution is weakly correlated across different tree branches

new exact solutions for polynomial oscillators in large dimensions a new type of exact solvability is reported the schrodinger equation is considered in a very large spatial dimension d 1 and its central polynomial potential is allowed to depend on many 2q coupling constants in a search for its bound states possessing an exact and elementary wavefunction proportional to a harmonic oscillator like polynomial of a freely varying i e not just small degree n the solvability conditions are known to form a complicated nonlinear set which requires a purely numerical treatment at a generic choice of d q and n assuming that d is large we discovered and demonstrate that this problem may be completely factorized and acquires an amazingly simple exact solution at all n and up to q 5 at least

large order perturbation expansions for the charged harmonic oscillator the charged oscillator defined by the hamiltonian h d2 dr2 r2 lambda r in the domain 0 infinity is a particular case of the family of spike oscillators which does not behave as a supersingular hamiltonian this problem is analysed around the three regions lambda to infinity lambda to 0 and lambda to infinity by using rayleigh ritz large order perturbative expansions a path is found to connect the large lambda regions with the small lambda region by means of the renormalization of the series expansions in lambda finally the riccati pade method is used to construct an implicit expansion around lambda to 0 which extends to very large values of mod lambda mod

singular continuous spectrum for the laplacian on certain sparse trees we present examples of rooted tree graphs for which the laplacian has singular continuous spectral measures for some of these examples we further establish fractional hausdorff dimensions the singular continuous components in these models have an interesting multiplicity structure the results are obtained via a decomposition of the laplacian into a direct sum of jacobi matrices

polynomial inverse images and polynomial inequalities valid for polynomials pn of degree at most n in this paper we are primarily interested in what form these inequalities take on several intervals we shall see that the extension to general sets involves the equilibrium measure of these sets we shall give the precise form of the bernstein inequality for arbi trary compacts and an asymptotical ly best form of the markoff inequality for sets consisting of finitely many intervals actually in this case we shall prove different markoff inequalities one one associated with each one of the endpoints of the system of intervals the proofs will heavily use sets that are obtained as the inverse images of intervals under special polynomial mappings we shall see that the original bernstein and

classification of oscillators in the heisenberg matrix representation for an arbitrary finite and not too singular presumably phenomenological superposition of potentials rdelta with rational exponents we solve the old problem of conversion of the corresponding differential schrodinger bound state problem into its matrix equivalent with the minimalized number l of non zero diagonals the construction with proofs is performed via the standard power series method in its non hermitean matrix sometimes called hill determinant version all the simplest and in this sense algebraizable interactions are then displayed up to l 5 an example of application in perturbation theory is included

a class of exact solutions for anharmonically coupled oscillators the author studies the solutions of the two dimensional schrodinger equation for anharmonically coupled oscillators the author obtains a finite set of exact solutions of the form psi x y polynomial x y exp polynomial x y provided that certain constraints on the potential parameters are satisfied

symmetry and approximability of submodular maximization problems a number of recent results on optimization problems involving submodular functions have made use of the multilinear relaxation of the problem these results hold typically in the value oracle model where the objective function is accessible via a black box returning f s for a given s we present a general approach to deriving inapproximability results in the value oracle model based on the notion of symmetry gap our main result is that for any fixed instance that exhibits a certain symmetry gap in its multilinear relaxation there is a naturally related class of instances for which a better approximation factor than the symmetry gap would require exponentially many oracle queries this unifies several known hardness results for submodular maximization e g the optimality of 1 1 e approximation for monotone submodular maximization under a cardinality constraint and the impossibility of frac12 epsilon approximation for unconstrained nonmonotone submodular maximization as a new applica

periodic jacobi matrices on trees abstract we begin the systematic study of the spectral theory of periodic jacobi matrices on trees including a formal definition the most significant result that appears here for the first time is that these operators have no singular continuous spectrum we review important previous results of sunada and aomoto and present several illuminating examples we present many open problems and conjectures that we hope will stimulate further work

graph cuts with interacting edge weights examples approximations and algorithms we study an extension of the classical graph cut problem wherein we replace the modular sum of edge weights cost function by a submodular set function defined over graph edges special cases of this problem have appeared in different applications in signal processing machine learning and computer vision in this paper we connect these applications via the generic formulation of cooperative graph cuts for which we study complexity algorithms and connections to polymatroidal network flows finally we compare the proposed algorithms empirically

study of the singular anharmonic potentials by means of the analytic continuation method here we study the singular anharmonic potentials by applying the analytic continuation method of holubec and stauffer 1985 in order to do that we have developed several approximations to the problem because this method cannot be applied when the solution has essential singularities at any point of its domain all the options here shown have the same precision giving us the eigenvalues correct to all the decimal places provided by the computer

analytic treatment of the αx2 βx4 oscillator we carry out an investigation of the schrodinger equation with the x2 x4 interaction for 0 re x and exp i in the sectors 0 arg x 3 6 0 and 6 arg x 3 6 0 a subdominant solution is constructed essentially in terms of the laplace transform of a function f s 0 s which is expressible as a converging power series in s the solution y x thus obtained is compared and contrasted with the one valid in the usual case where re x 0 is included and it is argued that y x may not be accessible to perturbative approaches

a basis for variational calculations in d dimensions in this paper we derive expressions for matrix elements i hj for the hamiltonian h qa q rq in d 2 dimensions the basis functions in each angular momentum subspace are of the form the matrix elements are given in terms of the gamma function for all d the significance of the parameters t and p and scale s are discussed applications to a variety of potentials are presented including potentials with singular repulsive terms of the form r 0 perturbed coulomb potentials d r br ar2 and potentials with weak repulsive terms such as r2 r4 0

extended continued fractions and energies of the anharmonic oscillators we describe the analytic solution to the schrodinger eigenvalue problem for the class of the central potentials v r za r where a 2 1 4 amax 0 z is an arbitrary finite set of the integer or rational exponents 2 1 0 the formalism is based on an analysis of the asymptotic behavior of the explicit regular solution r and issues in the formulation of the secular equation 1 l1 e 0 which determines the binding energies the main result is the rigorous construction of l1 e as a generalized extended and convergent continued fraction the method cannot be applied to the a i 1 0 cases this disproves the closely related hill determinant approach as conjectured recently by singh et al for the simplest potentials with z 2 2 4 6 and z 2 1 1 2

quantum states of a sextic potential hidden symmetry and quantum monodromy a known terminating polynomial ansatz for a finite sub set of states of the sextic central potential v a b c r ar2 br4 cr6 c 0 subject to integer related parameter constraints is confirmed to apply in two as well as three dimensions a hidden symmetry relating the eigenstates of v a b c r to those of v a b c r is used to establish the boundary between the ansatz sub set and the infinite set of higher states connections between the radial wavefunctions rn a b c r and rn a b c r also provide semiclassical insight into the origin of the parameter constraint finally the ansatz eigenvalue dispositions are related to the phenomenon of quantum monodromy

lower bounds for quartic anharmonic and double well potentials rigorous and remarkably accurate lower bounds to the lower eigenvalue spectrum of the schrodinger equation with quartic anharmonic and symmetric double well potentials of the form v a b ax2 2 bx4 b 0 are presented this procedure exploits some exactly soluble model potentials and appears to be of quite general utility

the rotation number for almost periodic potentials we define and analyze the rotation number for the almost periodic schrodinger operatorl d2 dx2 q x we use the rotation number to discuss i the spectrum ofl ii its relation to the korteweg de vries equation

exact solutions of the schrodinger equation for nonseparable anharmonic oscillator potentials in two dimensions an explicit procedure is given to construct exact closed form solutions to the time independent schrodinger equation in two dimensions 2 w x y 0 where w x y is a polynomial potential of degree greater than two not separable in cartesian coordinates several examples are discussed for which w x y is a sextic polynomial as has already been seen in studies of the corresponding one dimensional problem a complete set of eigenvalues and wave functions is not found however these closed form solutions can be used to check the accuracy and efficiency of numerical algorithms

transfer matrices hyperbolic geometry and absolutely continuous spectrum for some discrete schrodinger operators on graphs abstract we prove the existence of absolutely continuous spectrum for a class of discrete schrodinger operators on tree like graphs we consider potentials whose radial behaviour is subject only to an l bound in the transverse direction the potential must satisfy a condition such as periodicity the graphs we consider include binary trees and graphs obtained from a binary tree by adding edges possibly with weights our methods are motivated by the one dimensional transfer matrix method interpreted as a discrete dynamical system on the hyperbolic plane this is extended to more general graphs leading to a formula for the green s function bounds on the green s function then follow from the contraction properties of the transformations that arise in this generalization the bounds imply the existence of absolutely continuous spectrum

extended states in the anderson model on the bethe lattice abstract we prove that the anderson hamiltonianh von the bethe lattice has extended states for small disorder more precisely given any closed intervalicontained in the interior of the spectrum of the laplacian on the bethe lattice we prove that for small disorderh has purely absolutely continuous spectrum iniwith probability one i e ac h i iand pp h i sc h i with probability one and its integrated density of states is continuously differentiable on the intervali

further polynomial type eigenfunctions the method developed previously by the author for obtaining eigenfunctions in the form polynomial exponential polynomial for a linear second order differential equation in normal form is extended to embrace the general case when the interaction potential or square of the refractive index profile has the form polynomial polynomial various situations arise in the development of the theory some of which can be concluded algebraically but the majority require numerical calculations regarding the rank of quite a general matrix containing many unknown parameters

spiked harmonic oscillators a complete variational treatment is provided for a family of spiked harmonic oscillator hamiltonians h d2 dx2 bx2 x b 0 0 for arbitrary 0 a compact topological proof is presented that the set s n of known exact solutions for 2 constitutes an orthonormal basis of the hilbert space l2 0 closed form expressions are derived for the matrix elements of h with respect to s these analytical results and the inclusion of a further free parameter facilitate optimized variational estimation of the eigenvalues of h to high accuracy

singular anharmonicities and the analytic continued fractions ii the potentials v r ar2 br 4 cr 6 the c 0 results of paper i j math phys 30 23 1989 are extended in spite of the presence of an additional coupling constant the laurent series solutions of the schrodinger equation that are obtained remain similar to mathieu functions indeed the recurrences for coefficients preserve their three term character their analytic continued fraction solutions still converge etc the formulas become even slightly simpler for c 0 due to a certain symmetry of the equations to be solved an acceleration of convergence is better understood and a few numerical illustrations of efficiency are also delivered

kotani theory for one dimensional stochastic jacobi matrices we consider families of operators h on l2 given by h u n u n 1 u n 1 v n u n wherev is a stationary bounded ergodic sequence we prove analogs of kotani s results including that for a e ac h is the essential closure of the set ofe where e the lyaponov index vanishes and the result that ifv is non deterministic then ac is empty

maximizing stochastic monotone submodular functions we study the problem of maximizing a stochastic monotone submodular function with respect to a matroid constraint because of the presence of diminishing marginal values in real world problems our model can capture the effect of stochasticity in a wide range of applications we show that the adaptivity gap the ratio between the values of optimal adaptive and optimal nonadaptive policies is bounded and is equal to e e 1 we propose a polynomial time nonadaptive policy that achieves this bound we also present an adaptive myopic policy that obtains at least half of the optimal value furthermore when the matroid is uniform the myopic policy achieves the optimal approximation ratio of 1 1 e this paper was accepted by dimitris bertsimas and yinyu ye optimization

robust monotone submodular function maximization we consider a robust formulation introduced by krause et al j artif intell res 42 427 486 2011 of the classical cardinality constrained monotone submodular function maximization problem and give the first constant factor approximation results the robustness considered is w r t adversarial removal of up to tau elements from the chosen set for the fundamental case of tau 1 we give a deterministic 1 1 e 1 vartheta m approximation algorithm where m is an input parameter and number of queries scale as o n m 1 in the process we develop a deterministic 1 1 e 1 vartheta m approximate greedy algorithm for bi objective maximization of two monotone submodular functions generalizing the ideas and using a result from chekuri et al in focs 10 ieee pp 575 584 2010 we show a randomized 1 1 e epsilon approximation for constant tau and epsilon le frac 1 tilde varomega tau making o n 1 epsilon 3 queries further for tau ll sqrt k we give a fast and practical 0 387 algorithm finally we also give a black box result result for the much more general setting of robust maximization subject to an independence system

asymptotics of chebyshev polynomials i subsets of r we consider chebyshev polynomials t n z for infinite compact sets e that is the monic polynomials minimizing the sup norm t n e on e we resolve a 45 year old conjecture of widom that for finite gap subsets of r his conjectured asymptotics which we call szeg widom asymptotics holds we also prove the first upper bounds of the form t n e qc e n where c e is the logarithmic capacity of e for a class of e s with an infinite number of components explicitly for those e that obey a parreau widom condition

on the eigenvalues of the s state radial equation of a spiked harmonic oscillator eigenvalues of the ground state of the radial schrodinger equation for a spiked harmonic oscillator potential have been evaluated employing two methods numerically via the lanczos grid technique and by means of standard pade approximants constructed from an expansion of large coupling parameter series for the energy numerical results are compared for several values of the parameters characterizing the spiked singular potential

quantum electrodynamics of confined nonrelativistic particles abstract we consider a system of finitely many nonrelativistic quantum mechanical electrons bound to static nuclei the electrons are minimally coupled to the quantized electromagnetic field but we impose an ultraviolet cutoff on the electromagnetic vector potential appearing in covariant derivatives and the interactions between the radiation field and electrons localized very far from the nuclei are turned off for a class of hamiltonians we prove exponential localization of bound states establish the existence of a ground state and derive sufficient conditions for its uniqueness furthermore we show that excited bound states of the unperturbed system become unstable and turn into resonances when the electrons are coupled to the radiation field to this end we develop a novel renormalization transformation which acts directly on the space of hamiltonians

on the classical theory of ordinary linear differential equations of the second order and the schrodinger equation for power law potentials the power law potentials in the schrodinger equation solved recently are shown to come from the classical treatment of the singularities of a linear second order differential equation this allows one to enlarge the class of solvable power law potentials

spiked and symmetrized decadic potentials supporting elementary n plets of bound states we show that the potential wells v x x10 ax8 bx6 cx4 dx2 f x2 with a central spike possess arbitrary finite multiplets of elementary exact bound states their strong asymptotic growth implies an ambiguity in their symmetrically generalized quantization via complex boundary conditions but the three eligible methods coincide at our exceptional solutions

study of anharmonic singular potentials a simple and efficient variational method is introduced to accelerate the convergence of the eigenenergy computations for a hamiltonian h with singular potentials closed form analytic expressions in n dimensions are obtained for the matrix elements of h with respect to the eigenfunctions of a soluble singular problem with two free parameters a and b the matrix eigenvalues are then optimized with respect to a and b for a given n applications convergence rates and comparisons with earlier work are discussed in detail

orthogonal polynomials with a constant recursion formula and an application to harmonic analysis on calcule la mesure pour laquelle une suite de polynomes avec une formule de recurrence constante est orthogonale

on the applicability of the hill determinant and the analytic continued fraction method to anharmonic oscillators the authors finish a line of argument started in an earlier paper and show that the application of the hill determinant and the associated analytic continued fraction method for the calculation of eigenvalues of anharmonic oscillators is of dubious validity and may thus lead to erroneous results in this context some theorems concerning the eigenvalues of the lambda x4 and the doubly anharmonic oscillators are proved

on the αx2 βx4 interaction by applying the laplace transform method the authors obtain a class of exact solutions for the schrodinger equation with the potential v x alpha x2 beta x4 x 0 alpha or 0 beta 0 the corresponding energy spectrum being continuous a particular feature of these eigen solutions is that they may not be accessible to perturbation theory the properties of the solutions obtained are investigated and their relation to the case where x or 0 is included is discussed

absolutely continuous spectrum for the anderson model on a tree a geometric proof of klein s theorem we give a new proof of a version of klein s theorem on the existence of absolutely continuous spectrum for the anderson model on the bethe lattice at weak disorder

on the density of states of schrodinger operators with a random potential using very recent results on ergodic theorems for superadditive processes on rd the authors prove the existence of the density of states for a wide class of random schrodinger operators in particular new non asymptotic estimates on the density of states are obtained and examples are discussed

on the preservation of absolutely continuous spectrum for schrodinger operators abstract we present general principles for the preservation of a c spectrum under weak perturbations the schrodinger operators on the strip and on the caley tree bethe lattice are considered

the index of a pair of projections we discuss pairs of self adjoint projections with p q j 2n 1 the trace ideal and prove that for m n tr p q 2m 1 tr p q 2n 1 dim ker q ran p dim ker p ran q is an integer we also prove that there exists a unitary v interchanging p and q if and only if this integer is 0

an algebraic approach to problems with polynomial hamiltonians on euclidean spaces explicit expressions are given for the actions and radial matrix elements of basic radial observables on multi dimensional spaces in a continuous sequence of orthonormal bases for unitary su 1 1 irreps explicit expressions are also given for so n reduced matrix elements of basic orbital observables these developments make it possible to determine the matrix elements of polynomial and other hamiltonians analytically to within so n clebsch gordan coefficients and to select an optimal basis for a particular problem such that the expansion of eigenfunctions is most rapidly convergent

the generalized continued fractions and potentials of the lennard jones type for a broad class of the strongly singular potentials v r which are defined as superpositions of separate power law components the general solution of the corresponding schrodinger differential equation is constructed as an analog of mathieu functions the analogy is supported by the use of the generalized continued fractions the questions of convergence are analyzed in detail

orthogonal polynomials on several intervals via a polynomial mapping starting from a sequence pn x no of orthogonal polynomials with an orthogonality measure yurj supported on eo c 1 1 we construct a new sequence p x fi of orthogonal polynomials on t 1 eq t is a polynomial of degree tv with an orthogonality measure i that is related to no if eo 1 1 then e t 1 l l will in general consist of tv intervals we give explicit formulas relating pn x fi and pn x fio and show how the recurrence coefficients in the three term recurrence formulas for these orthogonal polynomials are related if one chooses t to be a chebyshev polynomial of the first kind then one gets sieved orthogonal polynomials

comparing apples and oranges query trade off in submodular maximization fast algorithms for submodular maximization problems have a vast potential use in applicative settings such as machine learning social networks and economics though fast algorithms were known for some special cases only recently such algorithms were considered in the general case of maximizing a monotone submodular function subject to a matroid independence constraint the known fast algorithm matches the best possible approximation guarantee while trying to reduce the number of value oracle queries the algorithm performs our main result is a new algorithm for this general case that establishes a surprising trade off between two seemingly unrelated quantities the number of value oracle queries and the number of matroid independence queries performed by the algorithm specifically one can decrease the former by increasing the latter and vice versa while maintaining the best possible approximation guarantee such a trade off is very useful since various applications might incur significantly diff

elementary bound states for the power law potentials the author shows how the exact polynomial exp polynomial bound state solutions to the radial schrodinger equation may be constructed for any potential of the form v r sigma delta in d gdelta rdelta where d is a finite set of arbitrary rational numbers and some of the couplings gdelta are not independent

on exact solutions of the schrodinger equation in the one and three dimensional schrodinger equation a broad class of the regular potentials rational functions of the rational powers of r may admit exact bound state solutions in the generalised harmonic oscillator elementary form psi r rsigma polynomial exp polynomial the necessary and sufficient conditions of this phenomenon are derived in the form of coupled algebraic equations the methods of their solution and a few examples are discussed in particular the well known coulombic and oscillator solvability and the similar recent results of singh et al 1978 and whitehead et al 1982 are reproduced as the two simplest special cases