polynomial bound on the number of scattering poles on montre que dans la diffusion par un potentiel infiniment differentiable a support compact les poles de l operateur de diffusion sont a distribution polynomiale

analytical solutions to a class of nonlinear schrodinger equations with cal pt like potentials we present closed form solutions to a certain class of one and two dimensional nonlinear schr odinger equations involving potentials with broken and unbroken pt symmetry in the one dimensional case these solutions are given in terms of jacobi elliptic functions hyperbolic and trigonometric functions some of these solutions are possible even when the corresponding pt symmetric potentials have a zero threshold in two dimensions hyperbolic secant type solutions are obtained for a nonlinear schr odinger equation with a nonseparable complex potential

identities and exponential bounds for transfer matrices this paper is about analytic properties of single transfer matrices originating from general block tridiagonal or banded matrices such matrices occur in various applications in physics and numerical analysis the eigenvalues of the transfer matrix describe localization of eigenstates and are linked to the spectrum of the block tridiagonal matrix by a determinantal identity if the block tridiagonal matrix is invertible it is shown that half of the singular values of the transfer matrix have a lower bound exponentially large in the length of the chain and the other half have an upper bound that is exponentially small this is a consequence of a theorem by demko moss and smith on the decay of matrix elements of the inverse of banded matrices this article is part of a special issue of journal of physics a mathematical and theoretical devoted to lyapunov analysis from dynamical systems theory to applications

two combinatorial models with identical statics yet different dynamics motivated by the problem of sorting we introduce two simple combinatorial models with distinct hamiltonians yet identical spectra and hence partition function and show that the local dynamics of these models is very different after a deep quench one model slowly relaxes to the sorted state whereas the other model becomes blocked by the presence of stable local minima

determinants of block tridiagonal matrices an identity is proven that evaluates the determinant of a block tridiagonal matrix with or without corners as the determinant of the associated transfer matrix or a submatrix of it

infinite toeplitz and laurent matrices with localized impurities this paper is concerned with the change of the spectra of infinite toeplitz and laurent matrices under perturbations in a prescribed finite set of sites the main result says that the spectrum of a toeplitz matrix with a non constant rational symbol is not affected by small localized impurities while such impurities can nevertheless enlarge the spectrum of the corresponding laurent matrix we also study the spectra that may emerge when randomly perturbing toeplitz or laurent matrices in a randomly chosen single site

solving satisfiability problems by fluctuations the dynamics of stochastic local search algorithms stochastic local search algorithms are frequently used to numerically solve hard combinatorial optimization or decision problems we give numerical and approximate analytical descriptions of the dynamics of such algorithms applied to random satisfiability problems we find two different dynamical regimes depending on the number of constraints per variable for low constraintness the problems are solved efficiently i e in linear time for higher constraintness the solution times become exponential we observe that the dynamical behavior is characterized by a fast equilibration and fluctuations around this equilibrium if the algorithm runs long enough an exponentially rare fluctuation towards a solution appears

transfer matrices non hermitian hamiltonians and resolvents some spectral identities i consider the n step transfer matrix t for a general block hamiltonian with eigenvalue equation where and are matrices and provide its explicit representation in terms of blocks of the resolvent of the hamiltonian matrix for the system of length n with boundary conditions i then introduce the related hamiltonian for the case and and provide an exact relation between the trace of its resolvent and together with an identity of thouless type connecting tr with the hamiltonian eigenvalues for the results are then extended to by showing that it is itself a transfer matrix besides being of mathematical interest the identities should be useful for an analytical approach in the study of spectral properties of a physically relevant class of transfer matrices

scattering theory and the trace of the wave group in 9 lax and phillips discuss the analogue for classical scattering theories in odd space dimensions of the poisson summation formula as generalized to compact manifolds by chazarain 3 colin de verdi e 141 and duistermaat and guillemin 5 and further generalized to manifolds with boundary in 1 71 however in 9 this trace formula is only proved for comparatively large values of the time variable see also ralston 14 1 bardos el al 2 this note is devoted to the proof of the same formula for all positive time for the case of scattering by a smooth compactly supported potential if n a is the number of poles a 3 then for every e 0 there exists c 0 such that

application of statistical mechanics to np complete problems in combinatorial optimisation recently developed techniques of the statistical mechanics of random systems are applied to the graph partitioning problem the averaged cost function is calculated and agrees well with numerical results the problem bears close resemblance to that of spin glasses the authors find a spin glass transition in the system and the low temperature phase space has an ultrametric structure this sheds light on the nature of hard computation problems

spectral value sets of closed linear operators we study how the spectrum of a closed linear operator on a complex banach space changes under affine perturbations of the form a a a d e here a d and e are given linear operators whereas is an unknown bounded linear operator that parametrizes the possibly unbounded perturbation d e the union of the spectra of the perturbed operators a with the norm of smaller than a given 0 is called the spectral value set of a at level in this paper we extend a known characterization of these sets for the matrix case to infinite dimensions and in so doing present a framework that allows for unbounded perturbations of closed linear operators on banach spaces the results will be illustrated by applying them to a delay system with uncertain parameters and to a partial differential equation with a perturbed boundary condition

asymptotics of resonances for a schrodinger operator with vector values abstract we show that the resonance counting function for a schrodinger operator in dimension one has an asymptotic expansion and calculate an explicit expression for the leading term in some situations

complex scaling and the distribution of scattering poles the purpose of this paper is to establish sharp polynomial bounds on the number of scattering poles for a general class of compactly supported self adjoint perturbations of the laplacian in r n odd we also consider more general types of bounds that give sharper estimates in certain situations the general conclusion can be stated as follows the order of growth of the poles is the same as the order of growth of eigenvalues for corresponding compact problems from the few known cases however the exact asymptotics are expected to be different the scattering poles for compactly supported perturbations were rigorously defined by lax and phillips 13 in a more general setting they correspond to resonances the study of which has a long tradition in mathematical physics in the lax phillips theory they appear as the poles of the meromorphic continuation of the scattering matrix and coincide with the poles of the meromorphic continuation of the resolvent of the perturbed operator because of the latter characterization they can be considered as the analogue of the discrete spectral data for problems on noncompact domains the problem of estimating the counting function

transfer matrices and tridiagonal block hamiltonians with periodic and scattering boundary conditions general hamiltonian matrices with tridiagonal block structure and the associated transfer matrices are investigated in the cases of periodic and scattering boundary conditions they arise from tight binding models with finite range hopping in one or more dimensions of space in the presence of a aharonov bohm flux or in multichannel scattering an identity relating the characteristic equation of the periodic hamiltonian with that of the transfer matrix is found allowing a detailed analysis of the bands a velocity matrix is defined with properties relevant for the band structure or for the channel structure in the scattering problem

relaxation in graph coloring and satisfiability problems using t 0 monte carlo simulation we study the relaxation of graph coloring k col and satisfiability k sat two hard problems that have recently been shown to possess a phase transition in solvability as a parameter is varied a change from exponentially fast to power law relaxation and a transition to freezing behavior are found these changes take place for smaller values of the parameter than the solvability transition results for the coloring problem for colorable and clustered graphs and for the fraction of persistent spins for satisfiability are also presented

random hermitian versus random non hermitian operators unexpected links we present some unexpected links between the spectral properties of ensembles of large hermitian and non hermitian random matrices within the formalism of free random variables we address the task of adding non hermitian random matrices to solve this problem we present a new approach 1 based on a generalization of the notion of the matrix valued green s function to the quaternion level and on an extension of free probability theory which allows us to introduce the setting of quaternion valued free probability theory we use this quaternion construction to solve the problem of a non hermitian random matrix of the type h1 ih2 with hermitian h1 h2 freely independent finally we mention conformal mapping relating spectral properties of hermitian h1 h2 to spectral properties of the non hermitian h1 ih2 model

random fibonacci sequences solutions to the random fibonacci recurrence xn 1 xn xn 1 decrease increase exponentially xn exp n for sufficiently small large in the limits 0 and we expand the lyapunov exponent in powers of and 1 respectively for the classical case of 1 we obtain exact non perturbative results in particular an invariant measure associated with ricatti variable rn xn 1 xn is shown to exhibit plateaux around all rational r

scattering asymptotics for riemann surfaces in this article we prove the optimal polynomial lower bound for the number of resonances of a surface with hyperbolic ends we also give weyl asymptotics for the relative scattering phase of such a surface the proofs are based on trace formulae analogous to those of the euclidean odd dimensional scattering the main technical ingredient is a new proof of the poisson formula theorem 5 7 which is applicable in the euclidean case as well our lower bound seems to be the first example of an optimal polynomial lower bound for the number of resonances holding for a general class of higher dimensional elliptic operators with no symmetries the previous general lower bounds or asymptotics were either nonoptimal 25 58 9 one dimensional or radial 65 67 and 54 41 1 or they required some degeneracy of the

relaxation and metastability in a local search procedure for the random satisfiability problem an analysis of the average properties of a local search procedure randomwalksat for the satisfaction of random boolean constraints is presented depending on the ratio alpha of constraints per variable reaching a solution takes a time t res growing linearly t res approximately tau res alpha n alpha alpha d with the size n of the instance the relaxation time tau res alpha in the linear phase is calculated through a systematic expansion scheme based on a quantum formulation of the evolution operator for alpha alpha d the system is trapped in some metastable state and resolution occurs from escape from this state through crossing of a large barrier an annealed calculation of the height zeta alpha of this barrier is proposed the polynomial to exponential cross over alpha d approximately 2 7 is not related to the onset of clustering among solutions occurring at alpha approximately 3 86

existence of resonances in magnetic scattering the schrodinger operator with a compactly supported magnetic field is shown to produce infinitely many resonances in any odd dimension 3 the proof is based on the poisson formula for resonances and properties of the magnetic heat invariants

statistical properties of eigenvectors in non hermitian gaussian random matrix ensembles statistical properties of eigenvectors in non hermitian random matrix ensembles are discussed with an emphasis on correlations between left and right eigenvectors two approaches are described one is an exact calculation for ginibre s ensemble in which each matrix element is an independent identically distributed gaussian complex random variable the other is a simpler calculation using n 1 as an expansion parameter where n is the rank of the random matrix this is applied to girko s ensemble consequences of eigenvector correlations which may be of physical importance in applications are also discussed it is shown that eigenvalues are much more sensitive to perturbations than in the corresponding hermitian random matrix ensembles it is also shown that in problems with time evolution governed by a non hermitian random matrix transients are controlled by eigenvector correlations

computation of growth rates of random sequences with multi step memory we extend the generating function approach to the computation of growth rate of random fibonacci sequences with long memory functional iteration equations are obtained and its general form is conjectured and proved based on which an asymptotic representation of the growth rate is obtained the validity of both the derived and the conjectured formula are verified upon comparison with monte carlo simulation a numerical scheme of the functional iteration is designed and implemented successfully

lower bounds on the number of scattering poles ii abstract we obtain lower bounds on the number of scattering poles for a class of abstract compactly supported perturbations of the laplacian in r n n odd they are applied to estimate the number of resonances for obstacle scattering and for hypoelliptic compactly supported perturbations of the laplacian the proof involves a development of lax phillips theory in a generalized setting a poisson formula for scattering poles and some simple tauberian arguments

products of random matrices for disordered systems products of random transfer matrices are applied to low dimensional disordered systems to evaluate numerically extensive quantities such as entropy and overlap probability distribution the main advantage is the possiblity to avoid numerical differentiation the method works for arbitrary disorder distributions at any temperature

distribution of poles for scattering on the real line abstract the density of scattering poles is shown to be proportional to the length of the convex hull of the support of the potential in the case of a potential with finite singularities at the endpoints of the support asymptotic formulae for the poles are given while in the c 0 case an example of a potential with infinitely many scattering poles on i r is constructed the scattering amplitude of a compactly supported potential is also characterized

spectral value sets a graphical tool for robustness analysis in this note we present a method to determine and visualize the set of all complex numbers to which at least one eigenvalue of a matrix a can be shifted by real or complex perturbations of the form a a a d e where d and e are fixed matrices and the unknown disturbance matrix satisfies 0 both real and complex perturbations are studied the graphical method may be considered as an extension of the classical root locus method to multivariable systems and multiparameter perturbations the results are illustrated by various examples

the spectra of large toeplitz band matrices with a randomly perturbed entry this paper is concerned with the union sp j k tn a of all possible spectra that may emerge when perturbing a large n n toeplitz band matrix tn a in the j k site by a number randomly chosen from some set the main results give descriptive bounds and in several interesting situations even provide complete identifications of the limit of sp j k tn a as n also discussed are the cases of small and large sets as well as the discontinuity of the infinite volume case which means that in general sp j k tn a does not converge to something close to sp j k tn a as n where t a is the corresponding infinite toeplitz matrix illustrations are provided for tridiagonal toeplitz matrices a notable special case

on large toeplitz band matrices with an uncertain block abstract this paper investigates the possible spectra of large finite dimensional toeplitz band matrices with perturbations impurities uncertainties in the upper left m m block the main result shows that the asymptotic spectrum of such a matrix is not affected by these perturbations provided they have sufficiently small norm this follows from analysis of structured pseudospectra structured spectral value sets in contrast for typical non hermitian toeplitz matrices there exist certain rank one perturbations of arbitrarily small norm that move an eigenvalue away from the asymptotic spectrum in the large dimensional limit

novel computation of the growth rate of generalized random fibonacci sequences a new formulation is proposed for the computation of average growth rates of generalized random fibonacci sequences based on the new formula a novel numerical scheme is designed and successfully implemented and interesting analytic asymptotic expansions are obtained for several examples

on the remarkable spectrum of a non hermitian random matrix model a non hermitian random matrix model proposed a few years ago has a remarkably intricate spectrum various attempts have been made to understand the spectrum but even its dimension is not known using the dyson schmidt equation we show that the spectrum consists of a non denumerable set of lines in the complex plane each line is the support of the spectrum of a periodic hamiltonian obtained by the infinite repetition of any finite sequence of the disorder variables our approach is based on the theory of words we make a complete study of all four letter words the spectrum is complicated because our matrix contains everything that will ever be written in the history of the universe including this particular paper

spectra pseudospectra and localization for random bidiagonal matrices there has been much recent interest initiated by work of the physicists hatano and nelson in the eigenvalues of certain random non hermitian periodic tridiagonal matrices and their bidiagonal limits these eigenvalues cluster along a bubble with wings in the complex plane and the corresponding eigenvectors are localized in the wings delocalized in the bubble here in addition to eigenvalues pseudospectra are analyzed making it possible to treat the nonperiodic analogues of these random matrix problems inside the bubble the resolvent norm grows exponentially with the dimension outside it grows subexponentially in a bounded region that is the spectrum of the infinite dimensional operator localization and delocalization correspond to resolvent matrices whose entries exponentially decrease or increase respectively with distance from the diagonal this article presents theorems that characterize the spectra pseudospectra and numerical range for the four cases of finite bidiagonal matrices infinite bidiagonal matrices stochastic toeplitz operators finite periodic matrices and doubly infinite bidiagonal matrices stochastic laurent operators c 2001 john wiley sons inc

spectral duality and distribution of exponents for transfer matrices of block tridiagonal hamiltonians i consider a general block tridiagonal matrix and the corresponding transfer matrix by allowing for a complex bloch parameter in the boundary conditions the two matrices are related by a spectral duality as a consequence i derive some analytic properties of the exponents of the transfer matrix in terms of the eigenvalues of the non hermitian block matrix some of them are the single matrix analogues of results holding for lyapunov exponents of an ensemble of block matrices which occur in models of transport the counting function of exponents is related to winding numbers of eigenvalues i discuss some implications of duality for the distribution real bands and complex arcs and the dynamics of eigenvalues

pseudospectra of linear operators if a matrix or linear operator a is far from normal its eigenvalues or more generally its spectrum may have little to do with its behavior as measured by quantities such as an or exp ta more may be learned by examining the sets in the complex plane known as the pseudospectra of a defined by level curves of the norm of the resolvent zi a 1 five years ago the author published a paper that presented computed pseudospectra of thirteen highly nonnormal matrices arising in various applications since that time analogous computations have been carried out for differential and integral operators this paper a companion to the earlier one presents ten examples each chosen to illustrate one or more mathematical or physical principles

regular spacings of complex eigenvalues in the one dimensional non hermitian anderson model we prove that in dimension one the non real eigenvalues of the non hermitian anderson nha model with a selfaveraging potential are regularly spaced the class of selfaveraging potentials which we introduce in this paper is very wide and in particular includes stationary potentials with probability one as well as all quasi periodic potentials it should be emphasized that our approach here is much simpler than the one we used before it allows us a to investigate the above mentioned spacings b to establish certain properties of the integrated density of states of the hermitian anderson models with selfaveraging potentials and c to obtain as a by product much simpler proofs of our previous results concerned with non real eigenvalues of the nha model

on the infinite dimensional qr algorithm spectral computations of infinite dimensional operators are notoriously difficult yet ubiquitous in the sciences indeed despite more than half a century of research it is still unknown which classes of operators allow for the computation of spectra and eigenvectors with convergence rates and error control recent progress in classifying the difficulty of spectral problems into complexity hierarchies has revealed that the most difficult spectral problems are so hard that one needs three limits in the computation and no convergence rates nor error control is possible this begs the question which classes of operators allow for computations with convergence rates and error control in this paper we address this basic question and the algorithm used is an infinite dimensional version of the qr algorithm indeed we generalise the qr algorithm to infinite dimensional operators we prove that not only is the algorithm executable on a finite machine but one can also recover the extremal parts of the spectrum and corresponding eigenvectors with convergence rates and error control this allows for new classification results in the hierarchy of computational problems that existing algorithms have not been able to capture the algorithm and convergence theorems are demonstrated on a wealth of examples with comparisons to standard approaches that are notorious for providing false solutions we also find that in some cases the iqr algorithm performs better than predicted by theory and make conjectures for future study

existence of resonances in three dimensions ifp is an elliptic self adjoint perturbation of the laplacian on 3 and the coefficients ofp decay super exponentially then we show thatp has infinitely many resonances the resonances are defined here as the poles of the meromorphic continuation of p 2 1

spectral properties of random non self adjoint matrices and operators i describe some numerical experiments which determine the degree of spectral instability of medium sized randomly generated matrices which are far from self adjoint the conclusion is that the eigenvalues are likely to be intrinsically uncomputable for similar matrices of a larger size i also describe a stochastic family of bounded operators in infinite dimensions for almost all of which the eigenvectors generate a dense linear subspace but the eigenvalues do not determine the spectrum my results imply that the spectrum of the non self adjoint anderson model changes suddenly on passing to the infinite volume limit

non hermitian localization and population biology the time evolution of spatial fluctuations in inhomogeneous d dimensional biological systems is analyzed a single species continuous growth model in which the population disperses via diffusion and convection is considered time independent environmental heterogeneities such as a random distribution of nutrients or sunlight are modeled by quenched disorder in the growth rate linearization of this model of population dynamics shows that the fastest growing localized state dominates in a time proportional to a power of the logarithm of the system size using an analogy with a schr odinger equation subject to a constant imaginary vector potential we propose a delocalization transition for the steady state of the nonlinear problem at a critical convection threshold separating localized and extended states in the limit of high convection velocity the linearized growth problem in d dimensions exhibits singular scaling behavior described by a d ensuremath 1 dimensional generalization of the noisy burgers equation with universal singularities in the density of states associated with disorder averaged eigenvalues near the band edge in the complex plane the burgers mapping leads to unusual transverse spreading of convecting delocalized populations

non hermitian localization and delocalization we study localization and delocalization in a class of non hermitian hamiltonians inspired by the problem of vortex pinning in superconductors in various simplified models we are able to obtain analytic descriptions in particular of the nonperturbative emergence of a forked structure the appearance of wings in the density of states we calculate how the localization length diverges at the localization delocalization transition we map some versions of this problem onto a random walker problem in two dimensions for a certain model we find an intricate structure in its density of states

the asymptotic spectra of banded toeplitz and quasi toeplitz matrices toeplitz matrices occur in many mathematical as well as scientific and engineering investigations this paper considers the spectra of banded toeplitz and quasi toeplitz matrices with emphasis on n

analytic continuation of resolvent kernels on noncompact symmetric spaces let x g k be a symmetric space of noncompact type and let be the laplacian associated with a g invariant metric on x we show that the resolvent kernel of admits a holomorphic extension to a riemann surface depending on the rank of the symmetric space this riemann surface is a branched cover of the complex plane with a certain part of the real axis removed it has a branching point at the bottom of the spectrum of it is further shown that this branching point is quadratic if the rank of x is odd and is logarithmic otherwise in case g has only one conjugacy class of cartan subalgebras the resolvent kernel extends to a holomorphic function on a branched cover of with the only branching point being the bottom of the spectrum

enumeration of simple random walks and tridiagonal matrices we present some old and new results in the enumeration of random walks in one dimension mostly developed in work on enumerative combinatorics the relation between the trace of the nth power of a tridiagonal matrix and the enumeration of weighted paths of n steps allows an easier combinatorial enumeration of paths it also seems promising for the theory of tridiagonal random matrices

distribution of eigenvalues in non hermitian anderson models we develop a theory which describes the behavior of eigenvalues of a class of one dimensional random non hermitian operators introduced recently by hatano and nelson we prove that the eigenvalues are distributed along a curve in the complex plane an equation for the curve is derived and the density of complex eigenvalues is found in terms of spectral characteristics of a reference hermitian disordered system the generic properties of the eigenvalue distribution are discussed

non hermitian tridiagonal random matrices and returns to the origin of a random walk we study a class of tridiagonal matrix models the q roots of unity models which includes the sign q 2 and the clock q models by feinberg and zee we find that the eigenvalue densities are bounded by and have the symmetries of the regular polygon with 2q sides in the complex plane furthermore the averaged traces of mk are integers that count closed random walks on the line such that each site is visited a number of times multiple of q we obtain an explicit evaluation for them

can spectral value sets of toeplitz band matrices jump abstract the spectral value set speb ca of a bounded linear operator a on l2 is the union of the spectra of all operators of the form a bkc where k

the centred travelling salesman at high temperature a recently formulated statistical mechanics method is used to study the phase transition occurring in a generalization of the travelling salesman problem tsp known as the centred tsp the method shows that the problem has clear signs of a crossover but is only able to access unscaled finite temperatures above the transition point the solution of the problem using this method displays a curious duality

the diffusion reaction d r hamiltonian and the solutions of certain types of linear and nonlinear d r equations in one dimension with a view to having further insight into the mathematical content of the non hermitian hamiltonian associated with the diffusion reaction d r equation in one dimension we investigate a the solitary wave solutions of certain types of its nonlinear versions and b the problem of real eigenvalue spectrum associated with its linear version or with this class of non hermitian hamiltonians for case a we use the standard techniques to handle the quadratic and cubic nonlinearities in the d r equation whereas for case b a newly proposed method based on an extended complex phase space is employed for a particular class of solutions an ermakov system of equations is also found for the linear case further corresponding to the classical version of the above one dimensional complex hamiltonian an equivalent integrable system of two two dimensional real hamiltonians is suggested

sharp bounds on the number of scattering poles for perturbations of the laplacian sharp bounds on the numbern r of the scattering poles in the disc z r for a large class of compactly supported perturbations not necessarily selfadjoint of the laplacian in n n 3 odd are obtained in particular in the elliptic case the estimaten r cr n c is proved