Time-independent perturbation theory with Lagrange multipliers

We derive the formulas for the energy and wavefunction of the time-independent Schrödinger equation with perturbation in a compact form. Unlike the conventional approaches based on Rayleigh–Schrödinger or Brillouin–Wigner perturbation theories, we employ a recently developed approach of matrix-valued Lagrange multipliers that regularizes an eigenproblem. The Lagrange-multiplier regularization makes the characteristic matrix for an eigenproblem invertible. After applying the constraint equation to recover the original equation, we find the solutions of the energy and wavefunction consistent with the conventional approaches. This formalism does not rely on an iterative way and the order-by-order corrections are easily obtained by taking the Taylor expansion. The Lagrange-multiplier regularization formalism for perturbation theory presented in this paper is completely new and can be extended to the degenerate perturbation theory in a straightforward manner. We expect that this new formalism is also pedagogically useful to give insights on the perturbation theory in quantum mechanics.

Lagrange-multiplier regularization of eigenproblem for Jx

We solve the eigenproblem of the angular momentum $$J_x$$ J x by directly dealing with the non-diagonal matrix unlike the conventional approach rotating the trivial eigenstates of $$J_z$$ J z . Characteristic matrix is reduced into a tri-diagonal form following Narducci–Orszag rescaling of the eigenvectors. A systematic reduction formalism with recurrence relations for determinants of any dimension greatly simplifies the computation of tri-diagonal matrices. Thus the secular determinant is intrinsically factorized to find the eigenvalues immediately. The reduction formalism is employed to find the adjugate of the characteristic matrix. Improving the recently introduced Lagrange-multiplier regularization, we identify that every column of that adjugate matrix is indeed the eigenvector. It is remarkable that the approach presented in this work is completely new and unique in that any information of $$J_z$$ J z is not required and only algebraic operations are involved. Collapsing of the large amount of determinant calculation with the recurrence relation has a wide variety of applications to other tri-diagonal matrices appearing in various fields. This new formalism should be pedagogically useful for treating the angular momentum problem that is central to quantum mechanics course.

Government size and economic growth in African emerging economies: does the BARS curve exist?

PurposeThis study is aimed at testing the validity of the BARS theory and determining the threshold level at which excessive government expenditure hampers growth. The data from 10 African emerging economies from 1988 to 2019 were used.Design/methodology/approachThe methodology comprises several different stages. In the first stage, an Lagrange Multiplier (LM) type test was employed to find the appropriate transition variable among all the candidate variables to assess the linearity between government expenditure and economic growth and to find the sequence for selecting the order m of the transition function. The linearity test helped to identify the nature of the relationships between government expenditure and economic growth. In the second stage, the model evaluation was tested using the wild cluster bootstrap-Lagrange Multiplier (WCL-LM) test to assess appropriateness of the model. Thirdly, the Panel smooth transition regression (PSTR) model with one regime was estimated to test the validity of the BARS curve.FindingsThe results revealed evidence of nonlinear effects between government expenditure and economic growth, where the size of the government spending was found to be a 27.84% share of GDP, above which government expenditure caused growth to decline in African emerging economies. The findings combined into an inverted U-shape relationship, in line with the BARS theory.Originality/valueThis study proposes that policy-makers ought to formulate prudent fiscal policies that encourage expenditure, which would improve growth for selected countries as their current level of spending is below the threshold. This might be done through: (1) a suitable investment portfolio and (2) spending more on infrastructure.

Consistent coupling of positions and rotations for embedding 1D Cosserat beams into 3D solid volumes

The present article proposes a mortar-type finite element formulation for consistently embedding curved, slender beams into 3D solid volumes. Following the fundamental kinematic assumption of undeformable cross-section s, the beams are identified as 1D Cosserat continua with pointwise six (translational and rotational) degrees of freedom describing the cross-section (centroid) position and orientation. A consistent 1D-3D coupling scheme for this problem type is proposed, requiring to enforce both positional and rotational constraints. Since Boltzmann continua exhibit no inherent rotational degrees of freedom, suitable definitions of orthonormal triads are investigated that are representative for the orientation of material directions within the 3D solid. While the rotation tensor defined by the polar decomposition of the deformation gradient appears as a natural choice and will even be demonstrated to represent these material directions in a $$L_2$$ L 2 -optimal manner, several alternative triad definitions are investigated. Such alternatives potentially allow for a more efficient numerical evaluation. Moreover, objective (i.e. frame-invariant) rotational coupling constraints between beam and solid orientations are formulated and enforced in a variationally consistent manner based on either a penalty potential or a Lagrange multiplier potential. Eventually, finite element discretization of the solid domain, the embedded beams, which are modeled on basis of the geometrically exact beam theory, and the Lagrange multiplier field associated with the coupling constraints results in an embedded mortar-type formulation for rotational and translational constraint enforcement denoted as full beam-to-solid volume coupling (BTS-FULL) scheme. Based on elementary numerical test cases, it is demonstrated that a consistent spatial convergence behavior can be achieved and potential locking effects can be avoided, if the proposed BTS-FULL scheme is combined with a suitable solid triad definition. Eventually, real-life engineering applications are considered to illustrate the importance of consistently coupling both translational and rotational degrees of freedom as well as the upscaling potential of the proposed formulation. This allows the investigation of complex mechanical systems such as fiber-reinforced composite materials, containing a large number of curved, slender fibers with arbitrary orientation embedded in a matrix material.