Two-sided eigenvalue bounds for the spherically symmetric states of the Schrödinger equation

AbstractThe eigenvalues of the radial Schrödinger equation are calculated very accurately by obtaining exact upper and lower bounds. By truncating the usual unbounded domain [0, ∞) of the system to a finite interval of the form [0,l], 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 double-well 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 l is in the neighborhood of some critical value, denoted by lcr

Accurate energy spectrum for double-well potential: periodic basis

We present a variational study of employing the trigonometric basis functions satisfying periodic boundary condition for the accurate calculation of eigenvalues and eigenfunctions of quartic double-well oscillators. Contrary to usual Dirichlet boundary condition, imposing periodic boundary condition on the basis functions results in the existence of an inflection point with vanishing curvature in the graph of the energy versus the domain of the variable. We show that this boundary condition results in a higher accuracy in comparison to Dirichlet boundary condition. This is due to the fact that the periodic basis functions are not necessarily forced to vanish at the boundaries and can properly fit themselves to the exact solutions.Comment: 15 pages, 5 figures, to appear in Molecular Physic

Variational collocation for systems of coupled anharmonic oscillators

We have applied a collocation approach to obtain the numerical solution to the stationary Schr\"odinger equation for systems of coupled oscillators. The dependence of the discretized Hamiltonian on scale and angle parameters is exploited to obtain optimal convergence to the exact results. A careful comparison with results taken from the literature is performed, showing the advantages of the present approach.Comment: 14 pages, 10 table

Assessing the transition of municipal solid waste management using combined material flow analysis and life cycle assessment

Faced with the challenges to deal with increasingly growing and ever diversified municipal solid waste (MSW), a series of waste directives have been published by European Commission to divert MSW from landfills to more sustainable management options. The presented study assessed the transition of MSW man-agement in Nottingham, UK, since the enforcement of the EU Landfill Directive using a tool of combined materials flow analysis (MFA) and life cycle assess-ment (LCA). The results show that the MSW management system in Nottingham changed from a relatively simple landfill & energy from waste (EfW) mode to a complex, multi-technology mode. Improvements in waste reduction, material re-cycling, energy recovery, and landfill prevention have been made. As a positive result, the global warming potential (GWP) of the MSW management system re-duced from 1,076.0 kg CO2–eq./t of MSW in 2001/02 to 211.3 kg CO2–eq./t of MSW in 2016/17. Based on the results of MFA and LCA, recommendations on separating food waste and textile at source and updating treatment technologies are made for future improvement

Novel Bound States Treatment of the Two Dimensional Schrodinger Equation with Pseudocentral Plus Multiparameter Noncentral Potential

By converting the rectangular basis potential V(x,y) into the form as V(r)+V(r, phi) described by the pseudo central plus noncentral potential, particular solutions of the two dimensional Schrodinger equation in plane-polar coordinates have been carried out through the analytic approaching technique of the Nikiforov and Uvarov (NUT). Both the exact bound state energy spectra and the corresponding bound state wavefunctions of the complete system are determined explicitly and in closed forms. Our presented results are identical to those of the previous works and they may also be useful for investigation and analysis of structural characteristics in a variety of quantum systemsComment: Published, 16 page

Pseudospectral methods for solving an equation of hypergeometric type with a perturbation

AbstractAlmost all, regular or singular, Sturm–Liouville eigenvalue problems in the Schrödinger form −Ψ″(x)+V(x)Ψ(x)=EΨ(x),x∈(ā,b̄)⊆R,Ψ(x)∈L2(ā,b̄) for a wide class of potentials V(x) may be transformed into the form σ(ξ)y″+τ(ξ)y′+Q(ξ)y=−λy,ξ∈(a,b)⊆R by means of intelligent transformations on both dependent and independent variables, where σ(ξ) and τ(ξ) are polynomials of degrees at most 2 and 1, respectively, and λ is a parameter. The last form is closely related to the equation of the hypergeometric type (EHT), in which Q(ξ) is identically zero. It will be called here the equation of hypergeometric type with a perturbation (EHTP). The function Q(ξ) may, therefore, be regarded as a perturbation. It is well known that the EHT has polynomial solutions of degree n for specific values of the parameter λ, i.e. λ:=λn(0)=−n[τ′+12(n−1)σ″], which form a basis for the Hilbert space L2(a,b) of square integrable functions. Pseudospectral methods based on this natural expansion basis are constructed to approximate the eigenvalues of EHTP, and hence the energies E of the original Schrödinger equation. Specimen computations are performed to support the convergence numerically

Singular inverse Sturm-Liouville problems with Hermite pseudospectral methods

A numerical approximation to recover certain symmetric potentials in the singular inverse Sturm-Liouville problems over (-infinity,infinity) is presented. A Hermite pseudospectral method is employed to cope with the corresponding direct problem on the real line, which is encountered in the iterative procedure proposed for the inverse problem. The usual but unwelcome ill-posed structure of the resulting numerical algorithm has been treated to some extent by the help of a flexible optimization parameter and regularization techniques. The construction of some specific potentials are illustrated to emphasize the effectiveness of the present scheme

A Fourier-Bessel Expansion for Solving Radial Schrödinger Equation in Two Dimensions

ABSTRACT The spectrum of the two-dimensional Schrödinger equation for polynomial oscillators bounded by infinitely high potentials, where the eigenvalue problem is defined on a finite interval r ∈ [0, L), is variationally studied. The wave function is expanded into a Fourier-Bessel series, and matrix elements in terms of integrals involving Bessel functions are evaluated analytically. Numerical results presented accurate to thirty digits show that, by the time L approaches a critical value, the low-lying state energies behave almost as if the potentials were unbounded. The method is applicable to multi-well oscillators as well

The Laguerre pseudospectral method for the reflection symmetric Hamiltonians on the real line

Hermite-Weber functions provide a natural expansion basis for the numerical treatment of the Schrodinger equation on the whole real line. For the reflection symmetric Hamiltonians, however, it is shown here that the transformation of the problem over the half line and use of a Laguerre basis is computationally much more efficient in a pseudospectral scheme