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

linear free resolutions and minimal multiplicity let s k x x be a polynomial ring over a field and let a4 a m be a finitely generated graded module in the most interesting case a4 is an ideal of s for a given natural number p there is a great interest in the question can m be generated by homogeneous elements of degree p no simple answer say in terms of the local cohomology of m is known but somewhat surprisingly the stronger question can the jth syzygy of m be generated by elements of degree qp j for all j 0 l n does admit a simple response the following is neither new nor difficult to prove though it seems not well known

a discontinuous galerkin finite element method for time dependent partial differential equations with higher order derivatives in this paper we develop a new discontinuous galerkin dg finite element method for solving time dependent partial differential equations pdes with higher order spatial derivatives unlike the traditional local discontinuous galerkin ldg method the method in this paper can be applied without introducing any auxiliary variables or rewriting the original equation into a larger system stability is ensured by a careful choice of interface numerical fluxes the method can be designed for quite general nonlinear pdes and we prove stability and give error estimates for a few representative classes of pdes up to fifth order numerical examples show that our scheme attains the optimal k 1 th order of accuracy when using piecewise k th degree polynomials under the condition that k 1 is greater than or equal to the order of the equation

on the diameter of an ideal abstract we begin the study of the notion of diameter of an ideal i s k x 1 x n an invariant measuring the distance between the minimal primes of i we provide large classes of hirsch ideals i e ideals satisfying diam i height i such as quadratic radical ideals such that s i is gorenstein and height i 4 or ideals admitting a square free complete intersection initial ideal

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

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

discontinuous galerkin methods for ordinary differential equations a class of galerkin methods derived from discontinuous piecewise polynomial spaces is analyzed for polynomials of degree k these methods lead to a family of one step schemes generating approximations up to order 2k 2 for the solution of an ordinary differential equation

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

initial complexes of prime ideals abstract the simplicial complex defined by any initial ideal of a prime ideal is pure and strongly connected this proves a conjecture of kredel and weispfenning 12 p 234

on formal local cohomology and connectedness let a denote an ideal of a local ring r m let m be a finitely generated r module there is a systematic study of the formal cohomology modules lim hmi m anm i z we analyze their r module structure the upper and lower vanishing and non vanishing in terms of intrinsic data of m and its functorial behavior these cohomology modules occur in relation to the formal completion of the punctured spectrum specr v m as a new cohomological data there is a description on the formal grade fgrade a m defined as the minimal non vanishing of the formal cohomology modules there are various exact sequences concerning the formal cohomology modules among them a mayer vietoris sequence for two ideals it applies to new connectedness results there are also relations to local cohomological dimensions

a posteriori error estimation for discontinuous galerkin solutions of hyperbolic problems abstract we analyze the spatial discretization errors associated with solutions of one dimensional hyperbolic conservation laws by discontinuous galerkin methods dgms in space we show that the leading term of the spatial discretization error with piecewise polynomial approximations of degree p is proportional to a radau polynomial of degree p 1 on each element we also prove that the local and global discretization errors are o x2 p 1 and o x2p 1 at the downwind point of each element this strong superconvergence enables us to show that local and global discretization errors converge as o xp 2 at the remaining roots of radau polynomial of degree p 1 on each element convergence of local and global discretization errors to the radau polynomial of degree p 1 also holds for smooth solutions as p these results are used to construct asymptotically correct a posteriori estimates of spatial discretization errors that are effective for linear and nonlinear conservation laws in regions where solutions are smooth

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

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

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

on superheight conditions for the affineness of open subsets in this paper we consider the open complement u of a hypersurface y v a in an affine scheme x we study the relations between the affineness of u the intersection of y with closed subschemes the property that every closed surface in u is affine the property that every analytic closed surface is stein and the superheight of a defining ideal a

square free grobner degenerations let i be a homogeneous ideal of s k x 1 ldots x n and let j be an initial ideal of i with respect to a term order we prove that if j is radical then the hilbert functions of the local cohomology modules supported at the homogeneous maximal ideal of s i and s j coincide in particular text depth s i text depth s j and text reg s i text reg s j

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

simplicial ideals 2 linear ideals and arithmetical rank abstract in the first part of this paper we study scrollers and linearly joined varieties scrollers were introduced in barile and morales 2004 bm4 linearly joined varieties are an extension of scrollers and were defined in eisenbud et al 2005 eghp there they proved that scrollers are defined by homogeneous ideals having a 2 linear resolution a particular class of varieties of important interest in classical geometry are cohen macaulay varieties of minimal degree they were classified geometrically by the successive contribution of del pezzo 1885 dp bertini 1907 b and xambo 1981 x and algebraically in barile and morales 2000 bm2 they appear naturally studying the fiber cone of a codimension two toric ideals morales 1995 m gimenez et al 1993 1999 gms1 gms2 barile and morales 1998 bm1 ha 2006 h ha and morales 2009 hm let s be a polynomial ring and i s a homogeneous ideal defining a sequence of linearly joined varieties we compute depth s i we prove that c v depth s i 1 where c v is the connectedness dimension of the algebraic set defined by i we characterize sets of generators of i and give an effective algorithm to find equations as an application we prove that ara i projdim s i in the case where v is a union of linear spaces in particular this applies to any square free monomial ideal having a 2 linear resolution in the case where v is a union of linear spaces the ideal i can be characterized by a tableau which is an extension of a ferrer or young tableau all these results are new and extend results in barile and morales 2004 bm4 eisenbud et al 2005 eghp

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

cohomological dimension lyubeznik numbers and connectedness in mixed characteristic abstract we establish a second vanishing theorem for local cohomology modules over regular rings of unramified mixed characteristic which relates the connectedness of the spectrum of a ring with the vanishing of local cohomology applying this and new results on the mixed characteristic lyubeznik numbers we further study connectedness properties of the spectra of a certain class of rings

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

an accurate asymptotic approximation and precise numerical solution of highly sensitive troesch s problem abstract troesch s problem is a two point boundary value problem that arises in the confinement of a plasma column by radiation pressure and in the theory of gas porous electrodes it has attracted significant amount of interest for the last 50 years current numerical solvers developed so far suffers significantly for large values of the sensitivity parameter here we present a new scheme that can be shown to outperform every alternative numerical evaluation procedure and can provide accurate solutions for arbitrarily large values of the parameter our solution based on derivation of a new asymptotic form where the difference of solution and this approximation decays exponentially for moderate values of the parameter further we use a discontinuous galerkin method to calculate this difference we include a variety of numerical results that confirm that indeed our algorithm compares favorably with alternative methods

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

a variational iteration method for solving troesch s problem troesch s problem is an inherently unstable two point boundary value problem a new and efficient algorithm based on the variational iteration method and variable transformation is proposed to solve troesch s problem the underlying idea of the method is to convert the hyperbolic type nonlinearity in the problem into polynomial type nonlinearities by variable transformation and the variational iteration method is then directly used to solve this transformed problem only the second order iterative solution is required to provide a highly accurate analytical solution as compared with those obtained by other analytical and numerical methods

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

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

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

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

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 discontinuous galerkin finite element method for solving the troesch s problem abstract in this paper we propose a new discontinuous galerkin finite element dg method to solve troesch s problem which is highly sensitive for large values of the parameter this two point boundary value problem has been heavily studied since 1960 however only a few papers have provided a reliable solution for high sensitivity therefore we developed the dg method which has proved its efficiency for many decades to be a new numerical solver we demonstrate through computational results compared with those computed by other methods that the discontinuous galerkin method provides a quite efficient accurate and reliable solution thus the dg method is an attractive and competitive alternative to other numerical and semi analytical techniques to solve highly sensitive nonlinear problems

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

shifted 1 n expansion approach to the interaction v r r2 λr2 1 gr2 the authors use the shifted 1 n expansion method to calculate the energy eigenvalues of the non polynomial oscillator v r r2 r2 1 gr2 the results thus obtained are in excellent agreement with exact numerical results it is also shown that the non polynomial interaction can be made supersymmetric and in this case also the results obtained by the shifted 1 n method matches extremely well with the exact analytic results given by supersymmetry

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

numerical solution of troesch s problem by simple shooting method this paper describes a simple and efficient approach to the troesch s problem in this approach the hyperbolic nonlinear term in the equation is first converted into polynomial nonlinear terms by variable transformation and a simple shooting method is then used directly to solve this transformed problem the calculated results are in excellent agreement with those obtained by other analytical and numerical methods

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

time discretization of parabolic problems by the hp version of the discontinuous galerkin finite element method the discontinuous galerkin finite element method dgfem for the time discretization of parabolic problems is analyzed in the context of the hp version of the galerkin method error bounds which are explicit in the time steps as well as in the approximation orders are derived and it is shown that the hp dgfem gives spectral convergence in problems with smooth time dependence in conjunction with geometric time partitions it is proved that the hp dgfem results in exponential rates of convergence for piecewise analytic solutions exhibiting singularities induced by incompatible initial data or piecewise analytic forcing terms for the h version dgfem algebraically graded time partitions are determined that give the optimal algebraic convergence rates a fully discrete hp scheme is discussed exemplarily for the heat equation the use of certain mesh design principles for the spatial discretizations yields exponential rates of convergence in time and space numerical examples confirm the theoretical results

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

grobner deformations connectedness and cohomological dimension abstract in this paper we will compare the connectivity dimension c p i of an ideal i in a polynomial ring p with that of any initial ideal of i generalizing a theorem of kalkbrener and sturmfels m kalkbrener b sturmfels initial complex of prime ideals adv math 116 1995 365 376 we prove that c p lt i min c p i dim p i 1 for each monomial order as a corollary we have that every initial complex of a cohen macaulay ideal is strongly connected our approach is based on the study of the cohomological dimension of an ideal a in a noetherian ring r and its relation with the connectivity dimension of r a in particular we prove a generalized version of a theorem of grothendieck a grothendieck cohomologie locale des faisceaux coherents et theoremes de lefschetz locaux et globaux sga 2 in seminaire de geometrie algebrique du bois marie 1962 as consequence of these results we obtain some necessary conditions for an open subscheme of a projective scheme to be affine

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

an efficient algorithm for solving troesch s problem abstract a new algorithm is presented for solving troesch s problem the numerical scheme based on the modified homotopy perturbation technique is deduced some numerical experiments are made compared with the variational iteration method and the adomian decomposition method the scheme is shown to be highly accurate and only a few terms are required to obtain accurate computable solutions finally the algorithm is applied to other problems

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

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

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 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

koszul property of projections of the veronese cubic surface abstract let v p 9 be the veronese cubic surface we classify the projections of v to p 8 whose coordinate rings are koszul in particular we obtain a purely theoretical proof of the koszulness of the pinched veronese a result obtained originally by caviglia using filtrations deformations and computer assisted computations to this purpose we extend to certain complete intersections results of conca herzog trung and valla concerning homological properties of diagonal algebras