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We describe a matrix diagonalization algorithm for complex symmetric (not Hermitian) matrices, A = A T , which is based on a two-step algorithm involving generalized Householder reflections based on the indefinite inner product 〈 u , v 〉 ∗ = ∑ i u i v i . This inner product is linear in both arguments and avoids complex conjugation. The complex symmetric input matrix is transformed to tridiagonal form using generalized Householder transformations (first step). An iterative, generalized QL decomposition of the tridiagonal matrix employing an implicit shift converges toward diagonal form (second step). The QL algorithm employs iterative deflation techniques when a machine-precision zero is encountered “prematurely” on the super-/sub-diagonal. The algorithm allows for a reliable and computationally efficient computation of resonance and antiresonance energies which emerge from complex-scaled Hamiltonians, and for the numerical determination of the real energy eigenvalues of pseudo-Hermitian and PT -symmetric Hamilton matrices. Numerical reference values are provided. Program summary Program Title: HTDQLS Program Files doi: http://dx.doi.org/10.17632/x24wjxtrsg.1 Licensing provisions: GPLv3 Programming language: Fortran 90 using fixed form notation Nature of problem: Calculating the eigenvalues and optionally the eigenvectors of complex symmetric (non-Hermitian), densely populated matrices. Solution method: The complex symmetric (not Hermitian) input matrix is diagonalized in two steps. First step: The matrix is tridiagonalized via a series of ( n − 2 ) generalized Householder reflections, where n is the rank of the input matrix. Second step: The tridiagonal matrix is diagonalized via a generalization of the “chasing the bulge” technique, which is an iterative process utilizing an implicitly shifted initial rotation followed by ( n − 2 ) Givens rotations. This technique is an implementation of QL factorization, and converges roughly as [ ( λ i − σ i ) ∕ ( λ i + 1 − σ i ) ] j where λ i is the eigenvalue located in the ( i , i ) position of the final diagonal matrix and the eigenvalues are ordered ( | λ 1 | | λ 2 | … | λ n | ), and j is the iteration. The “educated guess” σ i for the eigenvalue λ i is obtained from the analytic determination of the eigenvalues of ( k × k ) -submatrices of A , in the vicinity of the i th element, where k = 0 , 1 , 2 , 3 (here, k = 0 means that the implicit shift vanishes, σ i = 0 ). The routine optionally calculates the rotation matrix Z , such that Z − 1 A Z = D where A is the input matrix and D is the diagonal matrix containing the eigenvalues. The i th column of Z then is the eigenvector of A corresponding to the eigenvalue found at the element D ( i , i ) , in the i th position on the diagonal of the matrix D . Unusual features: For simplicity, the “wrapper” program which contains an example application and the HTDQLS routine are distributed in the same file.

An educational game for entrepreneurship was developed in a cooperation of Central University of Technology and Aalen University. The relation to Sustainable Development is twofold: the game will be a contribution to socio-economic development via entrepreneurship and a contribution to Education for Sustainable Development (ESD). In a project sponsored by the Baden-Wurttemberg Department of Science, Global Citizenship was also considered as an educational goal and was integrated into the game components.This paper discusses the role of educational games for entrepreneurial and managerial skills, sustainable development and global learning. We present the game system VAL-U based on the concept of value and of values. We analyse the concept development, the deployment project and first experiences with the rollout.

We investigate the coupling of the electromagnetic sources (charge and current densities) to the scalar and vector potentials in classical electrodynamics, using Green function techniques. As is well known, the scalar potential shows an action-at-a-distance behavior in Coulomb gauge. The conundrum generated by the instantaneous interaction has intrigued physicists for a long time. Starting from the differential equations that couple the sources to the potentials, we here show in a concise derivation, using the retarded Green function, how the instantaneous interaction cancels in the calculation of the electric field. The time derivative of a specific additional term in the vector potential, present only in Coulomb gauge, yields a supplementary contribution to the electric field which cancels the gradient of the instantaneous Coulomb gauge scalar potential, as required by gauge invariance. This completely eliminates the contribution of the instantaneous interaction from the electric field. It turns out that a careful formulation of the retarded Green function, inspired by field theory, is required in order to correctly treat boundary terms in partial integrations. Finally, compact integral representations are derived for the Lienard-Wiechert potentials (scalar and vector) in Coulomb gauge which manifestly contain two compensating action-at-a-distance terms., 12 pages; typographical error in Eq. (44a) corrected

This is the fourth paper in a series devoted to the large-order properties of anharmonic oscillators. We attempt to draw a connection of anharmonic oscillators to field theory, by investigating the partition function in the path integral representation around both the Gaussian saddle point, which determines the perturbative expansion of the eigenvalues, as well as the nontrivial instanton saddle point. The value of the classical action at the saddle point is the instanton action which determines the large-order properties of perturbation theory by a dispersion relation. In order to treat the perturbations about the instanton, one has to take into account the continuous symmetries broken by the instanton solution because they lead to zero-modes of the fluctuation operator of the instanton configuration. The problem is solved by changing variables in the path integral, taking the instanton parameters as integration variables (collective coordinates). The functional determinant (Faddeev–Popov determinant) of the change of variables implies nontrivial modifications of the one-loop and higher-loop corrections about the instanton configuration. These are evaluated and compared to exact WKB calculations. A specific cancellation mechanism for the first perturbation about the instanton, which has been conjectured for the sextic oscillator based on a nonperturbative generalized Bohr–Sommerfeld quantization condition, is verified by an analytic Feynman diagram calculation.

In view of the recently observed discrepancy of theory and experiment for muonic hydrogen [R. Pohl et al., Nature vol. 466, p. 213 (2010)], we reexamine the theory on which the quantum electrodynamic (QED) predictions are based. In particular, we update the theory of the 2P-2S Lamb shift, by calculating the self-energy of the bound muon in the full Coulomb+vacuum polarization (Uehling) potential. We also investigate the relativistic two-body corrections to the vacuum polarization shift, and we analyze the influence of the shape of the nuclear charge distribution on the proton radius determination. The uncertainty associated with the third Zemach moment < r^3 >_2 in the determination of the proton radius from the measurement is estimated. An updated theoretical prediction for the 2S-2P transition is given., 18 pages; LaTeX; literature references have been updated at the time of submission to a scientific journal

This is the third article in a series of three papers on the resonance energy levels of anharmonic oscillators. Whereas the first two papers mainly dealt with double-well potentials and modifications thereof [see J. Zinn-Justin and U. D. Jentschura, Ann. Phys. (N.Y.) 313 (2004), pp. 197 and 269], we here focus on simple even and odd anharmonic oscillators for arbitrary magnitude and complex phase of the coupling parameter. A unification is achieved by the use of PT-symmetry inspired dispersion relations and generalized quantization conditions that include instanton configurations. Higher-order formulas are provided for the oscillators of degrees 3 to 8, which lead to subleading corrections to the leading factorial growth of the perturbative coefficients describing the resonance energies. Numerical results are provided, and higher-order terms are found to be numerically significant. The resonances are described by generalized expansions involving intertwined non-analytic exponentials, logarithmic terms and power series. Finally, we summarize spectral properties and dispersion relations of anharmonic oscillators, and their interconnections. The purpose is to look at one of the classic problems of quantum theory from a new perspective, through which we gain systematic access to the phenomenologically significant higher-order terms.

Virtual Delbruck scattering in heavy-ion collisions contributes to the emission of photons. Specifically, the elastic scattering of an equivalent photon emitted by a nucleus in the Coulomb field of an incoming nucleus can lead to the emission of a photon in the collision of ultra-relativistic nuclei Z 1 Z 2 → Z 1 Z 2 γ . The discussed process has no infrared divergence. The total cross section obtained is 14 barn for the Au–Au collisions at the RHIC collider and 50 barn for the Pb–Pb collisions at the LHC collider. These cross sections are considerably larger than those for ordinary nuclear bremsstrahlung in the same photon energy range.

This review is focused on the borderline region of theoretical physics and mathematics. First, we describe numerical methods for the acceleration of the convergence of series. These provide a useful toolbox for theoretical physics which has hitherto not received the attention it actually deserves. The unifying concept for convergence acceleration methods is that in many cases, one can reach much faster convergence than by adding a particular series term by term. In some cases, it is even possible to use a divergent input series, together with a suitable sequence transformation, for the construction of numerical methods that can be applied to the calculation of special functions. This review both aims to provide some practical guidance as well as a groundwork for the study of specialized literature. As a second topic, we review some recent developments in the field of Borel resummation, which is generally recognized as one of the most versatile methods for the summation of factorially divergent (perturbation) series. Here, the focus is on algorithms which make optimal use of all information contained in a finite set of perturbative coefficients. The unifying concept for the various aspects of the Borel method investigated here is given by the singularities of the Borel transform, which introduce ambiguities from a mathematical point of view and lead to different possible physical interpretations. The two most important cases are: (i) the residues at the singularities correspond to the decay width of a resonance, and (ii) the presence of the singularities indicates the existence of nonperturbative contributions which cannot be accounted for on the basis of a Borel resummation and require generalizations toward resurgent expansions. Both of these cases are illustrated by examples., Comment: 119 pages, LaTeX

We analyze the effective action and the phase structure of N-layer sine-Gordon type models, generalizing the results obtained for the two-layer sine-Gordon model found in [I. Nandori, S. Nagy, K. Sailer and U. D. Jentschura, Nucl. Phys. B725, 467-492 (2005)]. Besides the obvious field theoretical interest, the layered sine-Gordon model has been used to describe the vortex properties of high transition temperature superconductors, and the extension of the previous analysis to a general N-layer model is necessary for a description of the critical behaviour of vortices in realistic multi-layer systems. The distinction of the Lagrangians in terms of mass eigenvalues is found to be the decisive parameter with respect to the phase structure of the N-layer models, with neighbouring layers being coupled by quadratic terms in the field variables. By a suitable rotation of the field variables, we identify the periodic modes (without explicit mass terms) in the N-layer structure, calculate the effective action and determine their Kosterlitz-Thouless type phase transitions to occur at a coupling parameter \beta^2_{c} = 8 N \pi, where N is the number of layers (or flavours in terms of the multi-flavour Schwinger model)., Comment: 15 pages

We discuss several applications of the recently proposed combined nonlinear-condensation transformation (CNCT) for the evaluation of slowly convergent, nonalternating series. These include certain statistical distributions which are of importance in linguistics, statistical-mechanics theory, and biophysics (statistical analysis of DNA sequences). We also discuss applications of the transformation in experimental mathematics, and we briefly expand on further applications in theoretical physics. Finally, we discuss a related Mathematica program for the computation of Lerch's transcendent.

We present a calculation of the B0-B0--mixing including Next--to--Leading Order (NLO) QCD corrections within the Two Higgs Doublet Model (2HDM). The QCD corrections at NLO are contained in the factor denoted by eta_2 which modifies the result obtained at the lowest order of perturbation theory. In the Standard Model case, we confirm the results for eta_2 obtained by Buras, Jamin and Weisz. The factor eta_2 is gauge and renormalization prescription invariant and it does not depend on the infrared behaviour of the theory, which constitutes an important test of the calculations. The NLO--calculations within the 2HDM enhance the LO--result up to 18%, which affects the correlation between M_H and V_{td}.

The decay rates of heavy atomic systems consisting of a particle and its own antiparticle are influenced by the probability density (“charge density”, square of the wave function) at the origin. In heavy systems, the one-loop electronic vacuum polarization induces a correction to the charge density (at the origin) of relative order α π . The correction is analyzed in this Letter, and results are provided for dimuonium ( μ + μ − atom), pionium ( π + π − atom), and the hypothetical heavy tauonium ( τ + τ − atom). We investigate the correction in the limit of strong binding and discuss the applicability of the effective charge approach to derive leading logarithmic asymptotics.

We report on the effects of growth temperature, arsenic source and post growth annealing treatment on the optical properties of low temperature grown GaAs. We have determined the absorption coefficient over a wide spectral range of energies between 0.8 and 2.0 eV. Special interest was dedicated to the shape of the band edge absorption for the different growth conditions and post growth treatments. The band edge is sharper for samples grown with an As 4 source than for samples grown using As 2 . However, the band edge becomes sharper due to the annealing process for all samples. A sharp band edge is essential for a high-contrast electro-optical modulator. We also measured the electro-absorption of a metal-semiconductor-metal structure based on annealed LT-GaAs and achieved a contrast ratio of 1:1.35 corresponding to an absorption change of 2300 cm −1 .