clustered model reduction of positive directed networks this paper proposes a clustered model reduction method for semistable positive linear systems evolving over directed networks in this method we construct a set of clusters i e disjoint sets of state variables based on a notion of cluster reducibility defined as the uncontrollability of local states by aggregating the reducible clusters with aggregation coefficients associated with the frobenius eigenvector we obtain an approximate model that preserves not only a network structure among clusters but also several fundamental properties such as semistability positivity and steady state characteristics furthermore it is found that the cluster reducibility can be characterized for semistable systems based on a projected controllability gramian that leads to an a priori h 2 error bound of the state discrepancy caused by aggregation the efficiency of the proposed method is demonstrated through an illustrative example of enzyme catalyzed reaction systems described by a chemical master equation this captures the time evolution of chemical reaction systems in terms of a set of ordinary differential equations

vectorial continued fractions and an algebraic construction of effective hamiltonians in the haromonic oscillator basis the effective hamiltonians are constructed for the polynomial interactions v r ti 1gir2i and t 4 their form factorized in terms of the vectorial continued fractions is simpler in a semirelativistic c reformulation its degeneracy in the c limit confluence of the physical and unphysical solutions is removed by solving a coupled set of the nonlinear algebraic equations in reduce

on the schrodinger equation for the interaction x2 λx2 1 gx2 the method of obtaining the explicit expressions of all the even odd parity exact solutions of the schrodinger equation for the interaction x2 lambda x2 1 gx2 is discussed when the couplings lambda and g satisfy some specific relations in the general case a simple equation for approximating the energy eigenvalues has been developed

fourier series for accurate stable reduced order models in large scale linear applications a new method fourier model reduction for obtaining stable accurate low order models of very large linear systems is presented the technique draws on traditional control and dynamical system concepts and utilizes them in a way which is computationally very efficient discrete time fourier coefficients of the large system are calculated and used to construct a reduced order model that preserves stability properties of the original system many coefficients can be calculated which results in a very accurate representation of the system dynamics but only a single factorization of the large system is required the resulting system can be further reduced using explicit formulae for balanced truncation the method is applied to two computational fluid dynamic systems which model unsteady motion of a two dimensional subsonic airfoil and unsteady flow in a supersonic diffuser in both cases the new method is found to work extremely well results are compared to models developed using the proper orthogonal decomposition and arnoldi method in comparison with these widely used techniques the new method is computationally more efficient preserves the stability of the original system uses both input and output information and for smooth transfer functions is valid over a wide range of frequencies

pairs of analytical eigenfunctions for the x2 λ x2 1 gx2 interaction a general method is discussed for deriving pairs of exact analytical eigenfunctions of the x2 x2 1 gx2 interaction recently discussed by blecher and leach and commented on by gallas it is shown that those pairs always consist of an odd and an even parity solution in each case and g are related by the conditions 0 and g as a result of their investigations the authors conjecture the possible existence of an infinite number of solution pairs having and g connected by several different relations

exact solutions of the harmonic oscillator plus non polynomial interaction the exact solutions to a one dimensional harmonic oscillator plus a non polynomial interaction ax2bx2 1cx2 a 0 c 0 are given by the confluent heun functions hc z the minimum value o

high precision calculation of the eigenvalues for the x2 λx2 1 gx2 potential an analytic continuation procedure using a taylor series is used to produce very accurate wavefunctions and eigenvalues for the one dimensional anharmonic oscillator governed by the potential v x x2 x2 1 gx2

a quantum quasi harmonic nonlinear oscillator with an isotonic term the properties of a nonlinear oscillator with an additional term kg x2 characterizing the isotonic oscillator are studied the nonlinearity affects to both the kinetic term and the potential and combines two nonlinearities associated to two parameters and kg in such a way that for 0 all the characteristics of the standard isotonic system are recovered the first part is devoted to the classical system and the second part to the quantum system this is a problem of quantization of a system with position dependent mass of the form m x 1 1 x2 with a dependent non polynomial rational potential and with an additional isotonic term the schrodinger equation is exactly solved and the kg dependent wave functions and bound state energies are explicitly obtained for both 0

a generalized morse asymmetric potential and multiplets of its non numerical exact bound states the solvable morse model with its asymmetric phenomenological potential v r a 1 exp mu r ralpha 2 is generalized via a computer assisted algebraic construction we show that for certain three component phenomenological potentials v r a 1 exp mu r ralpha 2 b 1 exp mu r rbeta 3 c 1 exp mu r rgamma 4 the closed and exact bound states may exist not only in singlets but also in doublets and triplets

a simplified fixed point perturbation theory ii an application to the first few pade potentials for pt i see ibid vol 20 p 907 1987 in the radial schrodinger equation for bound states the central potentials v r vcoulombb vshort range are considered with vshort range replaced by its p j p pade approximants p j or 1 the earlier method is completed and a few coefficients of the fixed point perturbation theory asymptotic expansions of the effective hamiltonians are evaluated in reduce for p or 3

the anharmonic oscillator problem a new algebraic solution the second order one dimensional or radial differential schrodinger equation with the potential v r mu r2 nu r4 may be re interpreted as a difference equation of the fourth order indeed the hamiltonian is a pentadiagonal matrix in the standard harmonic oscillator basis mod n n 0 1 thus the authors construct its four independent general solutions by purely algebraic means via expansions in powers of n 1 14 next preserving the analogy between difference and differential equations the physical wavefunctions n mod psi and their energies are determined by the matching of the jost and regular solutions finally using the simplest matching conditions at the origin the resulting numerical algorithm is demonstrated to be both stable and quickly convergent

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

exact solutions of the schrodinger equation d dx2 x2 λx2 1 gx2 ψ x eψ x the authors prove the existence of a class of exact eigenvalues and eigenfunctions of the schrodinger equation for the potential x2 x2 1 gx2 when certain algebraic relations between and g hold some of the properties of these solutions are discussed it is shown that in a certain sense they may be regarded as sturmians for the schrodinger equation with the potential x2 g g2x2

rd oscillators with arbitrary d 0 and perturbation expansions with sturmians in contrast to widespread belief the current rayleigh schrodinger perturbation theory may provide an easy description of double well oscillators and or of the strongly anharmonic forces with an arbitrary power law asymptotical growth one has only to work in a suitable sturmian basis the feasibility and numerical efficiency of the construction is illustrated on a few one dimensional one body examples

all optimal hankel norm approximations of linear multivariable systems and their l error bounds the problem of approximating a multivariable transfer function g s of mcmillan degree n by s of mcmillan degree k is considered a complete characterization of all approximations that minimize the hankel norm is derived the solution involves a characterization of all rational functions s f s that minimize where s has mcmillan degree k and f s is anticavisal the solution to the latter problem is via results on balanced realizations all pass functions and the inertia of matrices all in terms of the solutions to lyapunov equations it is then shown that where k 1 g s is the k l st hankel singular value of g s and for one class of optimal hankel norm approximations the method is not computationally demanding and is applied to a 12 state model

eigenvalues from power series expansions an alternative approach an appropriate rational approximation to the eigenfunction of the schrodinger equation for anharmonic oscillators enables one to obtain the eigenvalue accurately as the limit of a sequence of roots of hankel determinants the convergence rate of this approach is greater than that for a well established method based on power series expansions weighted by a gaussian factor with an adjustable parameter the so called hill determinant method

integral formulas for chebyshev polynomials and the error term of interpolatory quadrature formulae for analytic functions we evaluate explicitly the integrals 1 1 n t r t dt r 1 with the n being any one of the four chebyshev polynomials of degree n these integrals are subsequently used in order to obtain error bounds for interpolatory quadrature formulae with chebyshev abscissae when the function to be integrated is analytic in a domain containing 1 1 in its interior

potential v r μr2 νr4 and mutual equivalence of the two recent algebraic approaches to schrodinger equations the authors consider the anharmonic oscillator hamiltonian and solve the corresponding schrodinger equation by means of the recently proposed fixed point fp perturbation theory the fp formalism resembles an introduction of the creation and annihilation operators uses a vectorial generalisation of continued fractions introduces the inverse model space dimension as a natural and controllably small expansion parameter and is shown to define a smooth exact effective truncation of the hamiltonian matrix it also reproduces the recent n 1 asymptotic formulae for the wavefunctions

model reduction by moment matching for linear and nonlinear systems the model reduction problem for single input single output linear and nonlinear systems is addressed using the notion of moment a re visitation of the linear theory allows to obtain novel results for linear systems and to develop a nonlinear enhancement of the notion of moment this in turn is used to pose and solve the model reduction problem by moment matching for nonlinear systems to develop a notion of frequency response for nonlinear systems and to solve model reduction problems in the presence of constraints on the reduced model connections between the proposed results projection methods the covariance extension problem and interpolation theory are presented finally the theory is illustrated by means of simple worked out examples and case studies

singular anharmonicities and the analytic continued fractions the potential v r 2r2 2r 4 is investigated and its bound states are constructed by a generalized hill determinant method it is shown that the binding energy e and another free parameter in the underlying laurent series representation of the wave function may be obtained from a coupled pair of hill determinant equations

principal component analysis in linear systems controllability observability and model reduction kalman s minimal realization theory involves geometric objects controllable unobservable subspaces which are subject to structural instability specifically arbitrarily small perturbations in a model may cause a change in the dimensions of the associated subspaces this situation is manifested in computational difficulties which arise in attempts to apply textbook algorithms for computing a minimal realization structural instability associated with geometric theories is not unique to control it arises in the theory of linear equations as well in this setting the computational problems have been studied for decades and excellent tools have been developed for coping with the situation one of the main goals of this paper is to call attention to principal component analysis hotelling 1933 and an algorithm golub and reinsch 1970 for computing the singular value decompositon of a matrix together they form a powerful tool for coping with structural instability in dynamic systems as developed in this paper principal component analysis is a technique for analyzing signals singular value decomposition provides the computational machinery for this reason kalman s minimal realization theory is recast in terms of responses to injected signals application of the signal analysis to controllability and observability leads to a coordinate system in which the internally balanced model has special properties for asymptotically stable systems this yields working approximations of x c x bar o the controllable and unobservable subspaces it is proposed that a natural first step in model reduction is to apply the mechanics of minimal realization using these working subspaces

exact solutions for nonpolynomial potentials in n space dimensions using a factorization method and supersymmetry a supersymmetry inspired factorization method is used to obtain exact analytic solutions for one or a few quantum states of a broad class of nonpolynomial potentials in an arbitrary space dimension under suitable constraints on the potential parameters for the specific nonpolynomial oscillator potential v x x2 x2 1 gx2 previous analytic results including algebraic and integral type in one and three dimensions are easily obtainable from these compact expressions interesting aspects of the existence of pairs of solutions for certain combinations of and g are also discussed

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

dynamic group approach to the x2 λx2 1 gx2 potential a modified operator method based upon the so 2 1 dynamic group is applied to the one dimensional anharmonic oscillator potential v x x2 x2 1 gx2 a tilting transformation is carried out in order to improve the rate of convergence of the algebraic perturbation series very accurate results are obtained for the energy eigenvalues and for the wave functions especially in the case of small g values

on the hill determinant method the conditions of applicability of the hill determinant method to the potentials v r v1r v2r2 and v x x2 lambda x2 1 gx2 are investigated it is shown that the procedure leads to the actual eigenvalues in the former case provided the ansatz is properly chosen but it completely fails to yield the right answer in the latter one

extended numerov method for computing eigenvalues of specific schrodinger equations numerov s method and an extended version of it are introduced for computing eigenvalues of schrodinger equations with potentials v x which are even functions with respect to x furthermore it is assumed that the wavefunctions tend to zero for x to or infinity the derived results are compared with previously derived numerical data and with available exact values

fixed point perturbation theory and the potential r2 λr2 1 gr2 i analysis of convergence the simplest illustration of the recently suggested asymptotic perturbative approach to the band matrix hamiltonians is found in the harmonic oscillator complemented by the non polynomial anharmonicity lambda r2 1 gr2 in the paper the detailed construction of the effective hamiltonian is given and the convergence of its fixed point expansion is shown

error bounds and estimates for gauss turan quadrature formulae of analytic functions we study the error of gauss turan quadrature formulae when functions which are analytic on a neighborhood of the set of integration are considered a computable upper bound of the error is presented which is valid for arbitrary weight functions a comparison is made with the exact error and a number of numerical examples for arbitrary weight functions are given which show the advantages of using such rules as well as the sharpness of the error bound also a comparison is made with the other effective error bounds for some special weight functions appearing in the literature asymptotic error estimates when the number of nodes in the quadrature increases are presented a couple of numerical examples which show their efficiency are included

unified derivation of exact solutions for a class of quasi exactly solvable models we present a unified treatment of exact solutions for a class of four quantum mechanical models namely the anharmonic singular potential the generalized quantum isotonic oscillator the soft core coulomb potential and the non polynomially modified oscillator we show that all four cases are reducible to the same basic ordinary differential equation which is quasi exactly solvable a systematic and closed form solution to the basic equation is obtained via the bethe ansatz method using the result general exact expressions for the energies and the allowed potential parameters are given explicitly for each of the four cases in terms of the roots of a set of algebraic equations a hidden sl 2 algebraic structure is also discovered in these models

on the elementary schrodinger bound states and their multiplets the problem of the existence of elementary bound states is discussed a trivial observation that every elementary wave function i r is an exact bound state for an appropriate potential v r v i r r is shown to lead to a very transparent form of the quasiexact qe solvability condition v i v j for doublets and multiplets of the s in this sense the particular class of elementary ansatze i r r polynomial r2 exp r polynomial r2 also defines the particular class of qe solvable potentials they have an elementary nonpolynomial rational form possibly also with a strongly singular repulsive core at the origin the properties of these forces are discussed in detail

perturbed harmonic oscillator ladder operators eigenenergies and eigenfunctions for the x2 λx2 1 gx2 interaction the perturbed ladder operators method is applied to the resolution of the perturbed harmonic oscillator wave equation for the case where the perturbation is expandable in a convergent series of hermite polynomials hk when v x b2x2 sigma kckhk b12 x it is found that the use of a hermite polynomials basis together with the use of binomial coefficient functions in the quantum number greatly simplifies the determination of the perturbed ladder and factorisation functions thus one obtains analytical expressions of the eigenenergies and eigenfunctions up to any order of the perturbation without increasing intricacy thorough calculation has been given for a perturbing potential v x function even in x as an illustrative application of the procedure the resolution of the schrodinger equation with a potential function v x x2 lambda x2 1 gx2 g 0 is reinvestigated

the schrodinger equation for the interaction potential x2 λx2 1 gx2 and the first heun confluent equation the authors present systematically the solutions and the quasi polynomial solutions for the schrodinger equation with the x2 lambda x2 1 gx2 interaction potential with the help of the first heun confluent equation or spheroidal heun equation

linear system approximation via covariance equivalent realizations abstract a truncation technique for model reduction to simultaneously match a specified number of output covariance derivatives is described these covariance equivalent realizations which match q 1 covariance derivatives are called q covers in general q covers are not unique the additional freedom is used herein so that the q cover obtained also matches q markov parameters the truncation technique uses a form of the observability matrices of the full order system to determine a priori the order required of the reduced order model to match a specified number of output covariance derivatives and markov parameters the resulting realization is shown to be independent of the basis of the complete model to within a unitary transformation stability conditions for the reduced order model are also described and the relationships are established between stochastically equivalent realizations and the q covers

model reduction and clusterization of large scale bidirectional networks this paper proposes two model reduction methods for large scale bidirectional networks that fully utilize a network structure transformation implemented as positive tridiagonalization first we present a krylov based model reduction method that guarantees a specified error precision in terms of the h norm positive tridiagonalization allows us to derive an approximation error bound for the input to state model reduction without computationally expensive operations such as matrix factorization second we propose a novel model reduction method that preserves network topology among clusters i e node sets in this approach we introduce the notion of cluster uncontrollability based on positive tridiagonalization and then derive its theoretical relation to the approximation error this error analysis enables us to construct clusters that can be aggregated with a small approximation error the efficiency of both methods is verified through numerical examples including a large scale complex network

a finite difference approach for the calculation of perturbed oscillator energies a simple numerical method for calculating eigenvalues and corresponding eigenvectors of the schrodinger equation for a perturbed oscillator is described the derived results are compared with previously derived numerical data and with available exact values

potential r2 λr2 1 gr2 and the analytic continued fractions the author constructs the exact wavefunctions and an analytic continued fractional green function for the one and three dimensional schrodinger equation with the potential v r h r2 r2 lambda r2 1 gr2 h 1 4 lambda g 0 in the numerical application of the formulae the energy and norm of the boundary state psi may be approximated by both their lower and upper estimates

maximum of the modulus of kernels of gaussian quadrature formulae for one class of bernstein szegő weight functions abstract we continue with the study of the kernels k n z in the remainder terms r n f of the gaussian quadrature formulae for analytic functions f inside elliptical contours with foci at 1 and a sum of semi axes 1 the weight function w of bernstein szeg type here is w t w 1 2 t 1 1 t 2 1 4 1 2 t 2 1 t 1 1 1 0 sufficient conditions are found ensuring that the kernel attains its maximum absolute value at the intersection point of the contour with either the real or the imaginary axis this leads to effective error bounds of the corresponding gauss quadratures the quality of the derived bounds is demonstrated by a comparison with other error bounds intended for the same class of integrands

on the remainder term of gauss radau quadrature with chebyshev weight of the third kind for analytic functions abstract for analytic functions the remainder term of quadrature formulae can be represented as a contour integral with a complex kernel we study the kernel on elliptic contours with foci at the points 1 and a sum of semi axes 1 for gauss radau quadrature formula with chebyshev weight function of the third kind starting from the explicit expression of the corresponding kernel derived by gautschi we determine the locations on the ellipses where maximum modulus of the kernel is attained the obtained values confirm the corresponding conjectured values given by gautschi in his paper w gautschi on the remainder term for analytic functions of gauss lobatto and gauss radau quadratures rocky mounatin j math 21 1991 209 206 in this way the last unproved conjecture from the mentioned paper is now verified

model reduction of large scale systems by least squares abstract in this paper we introduce an approximation method for model reduction of large scale dynamical systems this is a projection which combines aspects of the svd and krylov based reduction methods this projection can be efficiently computed using tools from numerical analysis namely the rational krylov method for the krylov side of the projection and a low rank smith type iteration to solve a lyapunov equation for the svd side of the projection for discrete time systems the proposed approach is based on the least squares fit of the r 1 th column of a hankel matrix to the preceding r columns where r is the order of the reduced system the reduced system is asymptotically stable matches the first r markov parameters of the full order model and minimizes a weighted h 2 error the method is also generalized for moment matching at arbitrary interpolation points application to continuous time systems is achieved via the bilinear transformation numerical examples prove the effectiveness of the approach the proposed method is significant because it combines guaranteed stability and moment matching together with an optimization criterion

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

the remainder term for analytic functions of symmetric gaussian quadratures for analytic functions the remainder term of gaussian quadrature rules can be expressed as a contour integral with kernel k n in this paper the kernel is studied on elliptic contours for a great variety of symmetric weight functions including especially gegenbauer weight functions first a new series representation of the kernel is developed and analyzed then the location of the maximum modulus of the kernel on suitable ellipses is determined depending on the weight function the maximum modulus is attained at the intersection point of the ellipse with either the real or imaginary axis finally a detailed discussion for some special weight functions is given

comments on a paper concerning the analytic continued fraction technique for bound states the author shows that the treatment of bound and confined states for a class of confinement potentials by application of the analytic continued fraction method as presented in a recent paper dalta et al ibid vol 13 p 3161 1980 is erroneous and may lead to wrong results for the eigenvalues of the relevant schrodinger equation

a survey of model reduction by balanced truncation and some new results balanced truncation is one of the most common model reduction schemes in this note we present a survey of balancing related model reduction methods and their corresponding error norms and also introduce some new results five balancing methods are studied 1 lyapunov balancing 2 stochastic balancing 3 bounded real balancing 4 positive real balancing and 5 frequency weighted balancing for positive real balancing we introduce a multiplicative type error bound moreover for a certain subclass of positive real systems a modified positive real balancing scheme with an absolute error bound is proposed we also develop a new frequency weighted balanced reduction method with a simple bound on the error system based on the frequency domain representations of the system gramians two numerical examples are illustrated to verify the efficiency of the proposed methods

eigenenergies of the r2 lambda r2 1 gr2 and 1 2 r2 or gr4 1 g alpha r2 potentials obtained by the hypervirial method the energy levels of the schrodinger equation involving the potentials v r r2 lambda r2 1 gr2 and v or r 1 2 r2 or gr4 1 g alpha r2 are calculated by using hypervirial and pade approximant methods

an accurate finite difference method for the numerical solution of the schro dinger equation abstract an accurate finite difference approach for computing eigenvalues of schrodinger equations is developed in this paper we investigate two cases i the specific case in which the potential v x is an even function with respect to x it is assumed also that the wave functions tend to zero for x we investigate the well known potential of the onedimensional anharmonic oscillator the symmetric double well potential the razavy potential and the doubly anharmonic oscillator potential ii the general case for positive and negative eigenvalues and for the well known cases of the morse potential and woods saxon or optical potential numerical and theoretical results show that this new approach is more efficient than previously derived methods

brief paper krylov projection framework for fourier model reduction this paper analyzes the fourier model reduction fmr method from a rational krylov projection framework and shows how the fmr reduced model which has guaranteed stability and a global error bound can be computed in a numerically efficient and robust manner by monitoring the rank of the krylov subspace that underlies the fmr model the projection framework also provides an improved criterion for determining the number of fourier coefficients that are needed and hence the size of the resulting reduced order model the advantages of applying fmr in the rational krylov projection framework are demonstrated on a simple example

double exponential sinc collocation method for solving the energy eigenvalues of harmonic oscillators perturbed by a rational function we show that the double exponential sinc collocation method provides an efficient uniformly accurate solution to the one dimensional time independent schrodinger equation for a general class of rational potentials of the form v x p x q x the derived algorithm is based on the discretization of the hamiltonian of the schrodinger equation using sinc expansions this discretization results in a generalized eigenvalue problem the eigenvalues of which correspond to approximations of the energy values of the starting hamiltonian a systematic numerical study is conducted beginning with test potentials with known eigenvalues and moving to rational potentials of increasing degree

pade oscillators and a new formulation of perturbation theory being guided by the problem of bound states in potentials close to their pade approximants a new rayleigh schrodinger type perturbation theory is developed the unperturbed system is understood here in a broader sense its solutions are not needed but merely the related nondiagonal unperturbed propagator r in particular all the chain models h0 es0 h0 s0 band matrices with arbitrary perturbations are then perturbatively solvable with r constructed in terms of auxiliary matrix continued fraction fn alternatively a generalized unperturbed spectrum fn may be required as an input the algebraically constructed asymptotics of the fn s play this role in our pade examples due to s i the sturmians may also be constructed in the test evaluations of the binding energies and or couplings the simultaneous upper and lower bounds of high precision are shown to be numerically obtainable

accurate calculation of the eigenvalues of the x2 λx2 1 gx2 potential pour calculer les valeurs propres de h on calcule trois quantites qui convergent vers ces valeurs propres la difference entre 2 de ces quantites est une mesure de l erreur faite dans le calcul

error bounds for gauss turan quadrature formulae of analytic functions we study the kernels of the remainder term rn s f of gauss thran quadrature formulas 11f t w t dt ai vf i v rn s f n n s n0 for classes of analytic functions on elliptical contours with foci at 1 when the weight w is one of the special jacobi weights w t 1 t 1 t 1 2 1 2 s 1 2 1 2 s 1 2 s 1 2 we investigate the location on the contour where the modulus of the kernel attains its maximum value some numerical examples are included

a finite difference method for the numerical solution of the schro dinger equation a new approach which is based on a new property of phase lag for computing eigenvalues of schrodinger equations with potentials is developed in this paper we investigate two cases i the specific case in which the potential v x is an even function with respect to x it is assumed also that the wave functions tend to zero for x ii the general case of the morse potential and of the woods saxon or optical potential numerical and theoretical results show that this new approach is more efficient compared to previously derived methods

chain models and their effective hamiltonians we develop a new formalism suitable for diagonalization of hamiltonians with 2t 1 nonzero diagonals chain models in a systematic perturbation like way we get the expansions of green and or wave functions they are generated by solution of a sequence of fixed point quadratic equations for t t dimensional matrices for t 1 and t 2 this solution is feasible by elementary means so that the respective tridiagonal and pentadiagonal chain models may be considered exactly solvable in this context in an alternative first order variational perturbation formulation the method may provide the simultaneous upper and lower energy bounds for any t

error bounds for gaussian quadrature of analytic functions for gaussian quadrature rules over a finite interval applied to analytic or meromorphic functions we develop error bounds from contour integral representations of the remainder term as in previous work on the subject we consider both circular and elliptic contours in contrast with earlier work however we attempt to determine exactly where on the contour the kernel of the error functional attains its maximum modulus we succeed in answering this question for a large class of weight distributions including all jacobi weights when the contour is a circle in the more difficult case of elliptic contours we can settle the question for certain special jacobi weight distributions with parameters pm frac 1 2 and we provide empirical results for more general jacobi weights we further point out that the kernel of the error functional at any complex point outside the interval of integration can be evaluated accurately and efficiently by a recursive procedure the same procedure is useful also t

a simplified fixed point perturbation theory i the method for a simple potential the author proposes a conversion of the standard differential radial schrodinger equation into an infinite dimensional matrix equation a phi 0 and describes its systematic non numerical solution the results may be characterised as infinite asymptotic expansions of the wavefunctions phi and or of the secular determinant an inverse green function det a the inverse model space dimension plays a role of the small perturbation parameter

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

statistical mechanics of kinks for quasi exactly solvable potentials we study the thermodynamics of kinks in the presence of quasi exactly solvable qes potentials using a langevin code implemented on a massively parallel computer specifically we study the potential v 2 1 2 for which certain exact solutions of the schrodinger equation are known provided the parameters and satisfy certain constraints for 1 this potential has two degenerate minima otherwise it has only one minimum thus a system with this potential is capable of describing a second order phase transition a la the 4 model however no exact solutions of the schrodinger equation for this potential exist we have obtained an exact solution for the kink in this model in addition we have calculated in a 1d model such quantities as the probability density function pdf field configuration and field field correlation functions both above and below the transition temperature tc for several temperatures these quantities help us understand the contribution to the specific heat from coherent structures such as domain walls kinks as opposed to the contribution from lattice vibrations we have calibrated our results against known exact solutions for limiting cases with very high accuracy

a new method in the many body problem ii the earlier proposed method for exact determination of the energy spectra of hamiltonians h and the quantities where w is a finite dimensional perturbation is extended to treat a still larger class of hamiltonians in particular some two and three dimensional problems became amenable to this approach in addition the expressions for the density and local densities of states are deduced as an example an impurity problem is considered in a simple cubic lattice with short range and in a linear chain with long range interactions

study of a class of non polynomial oscillator potentials we develop a variational method to obtain accurate bounds for the eigenenergies of h v in arbitrary dimensions n 1 where v r is the nonpolynomial oscillator potential the variational bounds are compared with results previously obtained in the literature an infinite set of exact solutions is also obtained and used as a source of comparison eigenvalues

schrodinger equation as recurrences ii general solutions and their physical asymptotics for pt i see ibid vol 16 no 17 p 4001 11 1983 the general schrodinger equation is converted into equivalent 2t 1 term recurrences by the generalised lanczos construction of the basis then the exact bound states are given by the t parametric determinantal formula finally the values of the free parameters are shown to follow from the hill type normalisability requirement

on the x2 λx2 1 gx2 interaction presents an infinite set of exact solutions of odd parity for the one dimensional schrodinger equation with the interaction x2 lambda x2 1 gx2 these complement the exact solutions of even parity that have been obtained recently by flessas phys lett a vol 84 p 121 1981

model reduction for large scale dynamical systems via equality constrained least squares in this paper we present a new method of model reduction for large scale dynamical systems which belongs to the svd krylov based method category it is a two sided projection where one side reflects the krylov part and the other side reflects the svd observability gramian part the reduced model matches the first r i markov parameters of the full order model and the remaining ones approximate in a least squares sense without being explicitly computed where r is the order of the reduced system and i is a nonnegative integer such that email protected i

the error bounds of gauss radau quadrature formulae with bernstein szegő weight functions we consider the gauss radau quadrature formulae begin aligned int 1 1f t w t dt sum nu 1 n lambda nu f tau nu lambda n 1 f c r n f end aligned 11f t w t dt 1n f n 1f c rn f with c 1 c 1 or c 1 c 1 for the bernstein szeg weight functions consisting of anyone of the four chebyshev weights divided by the polynomial rho t 1 frac 4 gamma 1 gamma 2 t 2 t in 1 1 1 1 1 for the given quadrature formula starting from the explicit expression of the kernel we determine the locations on the ellipses where maximum modulus of the kernel is attained so we derive effective error bounds for this quadrature formula an alternative approach which has initiated this research has been proposed recently by notaris math comp 10 1090 mcom 2944 2015

harmonic oscillator well with a screened coulombic core is quasi exactly solvable in the quantization scheme which weakens the hermiticity of a hamiltonian to its mere t invariance the superposition v x x2 ze2 x of the harmonic and coulomb potentials is defined at the purely imaginary effective charges ze2 if and regularized by a purely imaginary shift of x this model is quasi exactly solvable we show that at each excited n 1 th harmonic oscillator energy e 2n 3 there exists not only the well known harmonic oscillator bound state at the vanishing charge f 0 but also a normalizable n 1 plet of the further elementary sturmian eigenstates n x at eigencharges f f n 0 n 0 1 n beyond the smallest multiplicities n we recommend perturbative methods for their construction

the error bounds of gauss quadrature formulae for the modified weight functions of chebyshev type abstract in this paper we consider the gauss quadrature formulae corresponding to some modifications of each of the four chebyshev weights considered by gautschi and li in 4 as it is well known in the case of analytic integrands the error of these quadrature formulas can be represented as a contour integral with a complex kernel we study the kernel of the mentioned quadrature formulas on suitable elliptic contours in such a way that the behavior of its modulus is analyzed in a rather simple manner allowing us to derive some effective error bounds in addition some numerical examples checking the accuracy of such error bounds are included

a new finite difference scheme with minimal phase lag for the numerical solution of the schrodinger equation a new finite difference approach for the numerical solution of the eigenvalue problem of the schrodinger equations is developed in this paper the new approach is based on the maximization of the algebraic order and the minimization of the phase lag we investigate two cases i the specific case in which the potential v x is an even function with respect to x it is assumed also that the wavefunctions tend to zero for x ii the general case for positive and negative eigenvalues and for the well known cases of the morse potential and woods saxon or optical potential numerical and theoretical results show that this new approach is more efficient when compared with previously derived methods

the quantum harmonic oscillator on the sphere and the hyperbolic plane κ dependent formalism polar coordinates and hypergeometric functions a nonlinear model representing the quantum harmonic oscillator on the sphere and the hyperbolic plane is solved in polar coordinates r by making use of a curvature dependent formalism the curvature is considered as a parameter and then the radial schrodinger equation becomes a dependent gauss hypergeometric equation the energy spectrum and the wave functions are exactly obtained in both the sphere s2 0 and the hyperbolic plane h2 0 a comparative study between the spherical and the hyperbolic quantum results is presented

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 a new approach has been used to show what kind of potentials can be used to obtain a series of exact solutions in closed forms of the schrodinger equation for bound states in particular the well known coulombic and oscillator solutions and others are reproduced as special cases

on computing eigenvalues of the schrodinger equation for symmetrical potentials the problem of the numerical determination of eigenvalues of the one dimensional schrodinger equation with symmetrical potential v x is considered the problem is reduced to the definition of an eigenvalue function g e well defined for the given potential v it is theoretically proved that the even parity eigenvalues are given by g e 0 and the odd parity eigenvalues are given by g e infinity the method is canonical in the sense that it is independent of the eigenfunction yet along with the eigenvalue it allows the determination of the eigenfunction initial values this method is applied to the potentials v x2 v x6 bx2 v x2 lambda x2 1 gx2 where exact eigenvalues are available the numerical results compared with the exact ones and with those of previous confirmed methods show that the present method is accurate and efficient

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