a hill series approach to wavefunction nodes the hill series method is shown to have several advantages for treating the schrodinger equation when the wavefunction obeys the dirichlet boundary condition r 0 a novel combination of the virial theorem the hill series method and a finite difference method is used to find the node positions of the wavefunction for several excited states of perturbed oscillator and coulombic systems and to find critical radii for the hydrogen atom

hypervirial perturbation theory revisited a wide range of topics in the area of hypervirial perturbation theory hvpt is discussed it is shown that with the use of a few simple procedures hvpt is capable of high accuracy for many problems results from many previous works in the literature are found to be improvable by the careful use of hvpt with appropriate choice of the unperturbed potential and of the origin at which the energy expansion is carried out two multi well problems from the literature are analysed in detail to show the value of a combination of hvpt and finite difference methods

study of discontinuities in eigenvalue spectra of some model hamiltonians in this paper the authors study the eigenvalue spectra of two model s wave hamiltonians in three dimensions h p2 2 1 r 2 mu r 2 lambda 2r2 and h p2 2 2 mu r 2 lambda 2r2 using the method of hill determinants they show that these eigenvalue spectra display discontinuities in the limit when both coupling constants mu and lambda vanish simultaneously they use a simple variational calculation to argue that such discontinuities are characteristic of these hamiltonians and are not an artefact of their numerical methods in fact such discontinuities are also to be seen in the case of a displaced harmonic oscillator in one dimension whose eigenvalue spectrum is exactly solvable

direct calculation of accurate siegert eigenvalues the roots of the hankel determinant built with the taylor coefficients of the regularized logarithmic derivative of the eigenfunction give both the bound state and siegert eigenvalues resonances of separable quantum mechanical systems we obtain accurate positions and widths for the resonances of some commonly chosen test examples

a solvable model of quantum discontinuities it is shown that an appropriate triple well version of the binding potential v x ax2 bx4 cx6 a c 0 b 0 b2 3ac provides a solvable model whereby the discontinuous behaviour of the energy eigenvalues of any finite set of lowest lying levels can be deduced analytically the reason for the occurrence of such discontinuities is discussed

a complex variable form of the heg technique a previously reported simple method for calculating complex matrix eigenvalues is modified to incorporate the traditional heg approach for the case of even parity potentials two examples of resonance calculations are given our matrix and perturbation results agree with each other but are not in full accord with previously published results for one of the test potentials new results are given for the resonances of the inverted gaussian potential

numerical quadrature rules for some infinite range integrals absrad recently the present author has given a new approach to numerical quadrature and derived new numerical quadrature formulas for finite range integrals with algebraic and or logarithmic endpoint singularities in the present work this approach is used to derive new numerical quadrature formulas for integrals of the form fl xae xf x dx and fl x ep x j x dx where ep x is the exponential integral it turns out the new rules are of interpolatory type their abscissas are distinct and lie in the interval of integration and their weights at least numerically are positive for fixed a the new integration rules have the same set of abscissas for all p finally the new rules seem to be at least as efficient as the corresponding gaussian quadrature formulas as an extension of the above numerical quadrature formulas for integrals of the form f i i x ie 2f x dx too are considered

biorthogonal polynomials and numerical quadrature formulas for some finite range integrals with symmetric weight functions in this work we derive a family of symmetric numerical quadrature formulas for finite range integrals i f email protected 1 1w x f x dx where w x is a symmetric weight function in particular we will treat the commonly occurring case of w x 1 x 2 a log 1 x 2 1 p p being a nonnegative integer these formulas are derived by applying a modification of the levin l transformation to some suitable asymptotic expansion of the function h z email protected 1 1w x z x dx as z and they turn out to be interpolatory the abscissas of these formulas have some rather interesting properties i they are the same for all a ii they are real and in 1 1 and iii they are related to the zeros of some known polynomials that are biorthogonal to certain powers of log 1 x 2 1 we provide tables and numerical examples that illustrate the effectiveness of our numerical quadrature formulas

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

the asymptotic iteration method for the eigenenergies of the schrodinger equation with the potential v r z r gr λr2 the eigenenergies of the schrodinger equation with coulomb potential plus linear and harmonic radial terms are investigated using the perturbation technique within the framework of the asymptotic iteration method the obtained eigenenergies are compared with those obtained from the shifted 1 n expansion method the moment method and also with the exact values obtained from the super symmetric method it is found that asymptotically this method gives accurate results for arbitrary g z and

comment on the asymptotic iteration method for polynomial potentials it is shown that polynomial potentials can be treated by a method which cuts out most of the complications of the popular aim approach several examples are given to show how expectation values and resonance energies can be calculated using the simplified approach

a simple method for complex eigenvalues a simple iterative method is described for finding the eigenvalues of a general square complex matrix several numerical examples involving complex symmetric matrices are treated in particular it is found that a naive matrix calculation without complex rotation produces resonant state energies in accord with those given by the recently introduced naive complex hypervirial perturbation theory

on the hill determinant method explicit examples are given of the false eigenvalues of the hill determinant method the existence of which has been suspected by workers using analytic methods a simple criterion is discovered for distinguishing between false and physical eigenvalues

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

two sided eigenvalue bounds for the spherically symmetric states of the schro dinger equation the eigenvalues of the radial schr6dinger equation are calculated very accurately by obtaining exact upper and lower bounds by truncating the usual unbounded domain 0 cx of the system to a finite interval of the form 0 g two auxiliary eigenvalue problems are defined it is then proved that the eigenvalues of the resulting confined systems provide upper and lower bounds converging monotonically to the true eigenvalues required moreover each auxiliary eigenvalue problem gives rise to an orthonormal set involving bessel functions the matrix representation of the hamiltonian is therefore derived by expanding the wave function into a fourier bessel series numerical results for single and doublewell polynomial oscillators as well as gaussian type non polynomial potentials illustrate that the eigenvalues can be calculated to an arbitrary accuracy whenever the boundary parameter g is in the neighborhood of some critical value denoted by ecr 1998 elsevier science b v all rights reserved

some recent results on the zeros of bessel functions and orthogonal polynomials several recent results are encountered on the zeros of bessel functions and of the classical orthogonal polynomials many of them are waiting for being published four problems are formulated as conjectures

direct expectation value calculations a new application of the hellmann feynman theorem is shown to allow accurate calculation of expectation values along with energies for the schrodinger equation for variable boundary position and boundary conditions power series and finite difference versions of the method are developed and applied to perturbed oscillator and perturbed hydrogen atom problems

an example of a quantum catastrophe the authors study the excited stage energies of the s wave hydrogen atom with the polynomial perturbation 2 lambda r 2 lambda 2r2 they demonstrate the existence of a discontinuity in the eigenvalue spectrum at lambda 0 as lambda changes from positive to negative

positive interpolatory quadrature rules generated by some biorthogonal polynomials interpolatory quadrature rules whose abscissas are zeros of a biorthogonal polynomial have proved to be useful especially in numerical integration of singular integrands however the positivity of their weights has remained an open question in some cases since 1980 we present a general criterion for this positivity as a consequence we establish positivity of the weights in a quadrature rule introduced by the second author in 1980 generated by a polynomial that is biorthogonal to log x j 0 j n 1

on eigenfunction approximations for typical non self adjoint schrodinger operators efficient approximations are constructed for the eigenfunctions of non self adjoint schrodinger operators in one dimension the same ideas also apply to the study of resonances of self adjoint schrodinger operators which have dilation analytic potentials in spite of the fact that such eigenfunctions can have surprisingly complicated structures with multiple local maxima we show that a suitable adaptation of the jwkb method is able to provide accurate global approximations to them

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

hill determinants and 1 n theory a computationally efficient form of the hill determinant method is devised giving both e and x2 values and is used to test the accuracy of that method in the light of recent criticisms l 1 and n 1 asymptotic theories are used to check the numerical results

on the applicability of the hill determinant method the authors show that contrarily to what flessas and anagnostatos claim 1982 the so called hill determinant method is perfectly suited to the numerical resolution of schrodinger s equation

complex variable hvpt complex variable hypervirial perturbation theory is applied to the case of oscillator and coulomb potentials perturbed by a single term potential of the form vxn or vrn respectively the trial calculations reported show that this approach can produce accurate complex energies for resonant states via a simple and speedy calculation and can also be useful in studies of pt symmetry and tunnelling resonance effects

a hill series method for resonances a simple algorithm is given to treat perturbed oscillator bound states and resonances the method is applied to a bound state problem for which the energy is a non analytic function of a perturbation parameter and also to two resonance problems one of which has a spectrum with an unusual dual nature

zeros of ultraspherical polynomials and the hilbert klein formulas abstract the orthogonality of the ultraspherical polynomials cn z for 1 2 ensures that all of their zeros are in the interval 1 1 in a previous paper driver and duren indag math 11 2000 43 51 we have shown that when 1 2 to 1 n in particular the pattern of migration from the interval 1 1 to the imaginary axis serves to confirm and explain the classical formulas of hilbert and klein for the number of zeros of cn z lying in each of the real intervals 1 1 1 and 1

the eigenvalues of the schrodinger equation for spherically symmetric states for various types of potentials in two three and n dimensions by using perturbative and non perturbative methods the energy levels of the schrodinger equation involving various model potentials in two three and n dimensional space are calculated by using perturbative and non perturbative methods the methods have been used effectively to calculate the energy eigenvalues even for high state numbers and large values of the perturbation parameters

numerical study of transport properties in continuum percolation the authors present numerical simulations of ac conductance for a random resistor capacitor network the conductance obeys a probability density function p g varies as g alpha 0 alpha 1 they use a highly efficient propagation algorithm to calculate the effective conductance of a long strip of a lattice at low frequencies they find that for the concentration p of conducting bonds less than the percolation threshold pc the imaginary part of conductance is proportional to frequency im geff approximately omega and the real part of conductance shows an anomalous frequency dependence re geff approximately omega 2 alpha the results of simulations in such a continuum system are in agreement with the predictions of the effective medium and the maxwell garnett approximation they also calculate the non universal dc conductivity exponents in continuum percolation the results are consistent with earlier theoretical predictions and numerical calculations

on the zeros of some polynomials that arise in numerical quadrature and convergence acceleration in this work the zeros of a sequence of polynomials that arise in convergence acceleration and some new numerical quadrature formulas are studied in particular it is proved that those polynomials that arise in numerical quadrature have all their zeros on 0 1 and that they are simple and a characterization theorem for these polynomials is also provided furthermore the zeros are shown to have an interlacing property

the exponential cosine screened potential the concept of forward nested multiplication is used to develop a unified power series hill determinant method for the calculation of schrodinger equation energy levels a numerical application is made to the exponential cosine screened potential

the hill determinant method in application to the sextic oscillator limitations and improvement using the sextic anharmonic oscillator a chi 2 b chi 4 c chi 6 as a testing ground it is shown why the hill determinant method has a limited range of applicability a possible improvement is suggested and discussed

on the improvement of convergence of hill determinants we propose a simple approach to analytically determine a parameter of a convergence factor exp gx2 4 in the hill determinant method it provides much more rapid convergence of the method than the other analytical approaches currently used for the g determination

false eigenvalues of the hill determinant method it is shown analytically that the false eigenenergies of a potential obtained by the method of hill determinants are the physical energies of a different potential

a connection between ultraspherical and pseudo ultraspherical polynomials abstract the pseudo ultraspherical polynomial of degree n is defined by c n x i n c n i x where c n x is the ultraspherical polynomial we derive a simple expression linking the polynomials c n and c n 1 2 n and show how to derive various properties of the zeros of c n when 1 n from properties of the zeros of c n when 1 2

asymptotic iteration method for eigenvalue problems an asymptotic iteration method for solving second order homogeneous linear differential equations of the form y 0 x y s0 x y is introduced where 0 x 0 and s0 x are c functions applications to schrodinger type problems including some with highly singular potentials are presented