Wolfgang Schweiger

This paper deals with the partial solution of the energy-eigenvalue problem for one-dimensional Schrödinger operators of the form $$H_N=X_0^2+V_N$$ H N = X 0 2 + V N , where $$V_N=X_N^2+\alpha X_{N-1}$$ V N = X N 2 + α X N - 1 is a polynomial potential of degree $$(2N-2)$$ ( 2 N - 2 ) and $$X_i$$ X i are the generators of an irreducible representation of a particular nilpotent group $${\mathcal{G}}_N$$ G N . Algebraization of the eigenvalue problem is achieved for eigenfunctions of the form $$\sum _{k=0}^M a_k X_2^k \exp (-\int dx\, X_N)$$ ∑ k = 0 M a k X 2 k exp ( - ∫ d x X N ) . It is shown that the overdetermined linear system of equations for the coefficients $$a_k$$ a k has a nontrivial solution, if the parameter $$\alpha$$ α and $$(N-3)$$ ( N - 3 ) Casimir invariants satisfy certain constraints. This general setting works for even $$N\ge 2$$ N ≥ 2 and can also be applied to odd $$N\ge 3$$ N ≥ 3 , if the potential is symmetrized by considering it as function of |x| rather than x. It provides a unified approach to quasi-exactly solvable polynomial interactions, including the harmonic oscillator, and extends corresponding results known from the literature. Explicit expressions for energy eigenvalues and eigenfunctions are given for the quasi-exactly solvable sextic, octic and decatic potentials. The case of $$E=0$$ E = 0 solutions for general N and M is also discussed. As physical application, the movement of a charged particle in an electromagnetic field of pertinent polynomial form is shortly sketched.

This paper deals with the partial solution of the energy eigenvalue problem for generalized symmetric quartic oscillators. Algebraization of the problem is achieved by expressing the Schrödinger operator in terms of the generators of a nilpotent group, which we call the quartic group. Energy eigenvalues are then seen to depend on the values of the two Casimir operators of the group. This dependence exhibits a scaling law which follows from the scaling properties of the group generators. Demanding that the potential gives rise to polynomial solutions in a particular Lie algebra element puts constraints on the four potential parameters, leaving only two of them free. For potentials satisfying such constraints, at least one of the energy eigenvalues and the corresponding eigenfunctions can be obtained in closed analytic form by pure algebraic means. With our approach, we extend the class of quasi-exactly solvable quartic oscillators which have been obtained in the literature by means of the more common $$\textrm{sl}(2,{\mathbb {R}})$$sl ( 2 , R ) algebraization. Finally, we show how solutions of the generalized quartic oscillator problem give rise to solutions for a charged particle moving in particular non-constant electromagnetic fields.

This paper deals with the description of weak $B^{-}\to D^0,{\pi }^0$ and $D^{-}\to K^0,{\pi }^0$ transition form factors in both the space- and timelike momentum transfer regions, within a constituent-quark model. To this aim, neutrino-meson scattering and semileptonic weak decays are formulated within the framework of point-form relativistic quantum mechanics to end up with relativistic invariant process amplitudes from which meson transition currents and form factors are extracted in an unambiguous way. For spacelike momentum transfers, these form factors depend on the frame in which the $WM{M}^{\prime }$ vertex is considered. On physical grounds, such a frame dependence is expected from a pure valence-quark picture, since a complete, frame independent description of form factors is supposed to require valence as well as nonvalence contributions. Nonvalence contributions, the most important being the $Z$ graphs, are, however, suppressed in the infinite-momentum frame ($q^2<0$). On the other hand, they can play a significant role in the Breit frame ($q^2<0$) and in the direct decay calculation ($q^2>0$), as a comparison with the infinite-momentum-frame form factors (analytically continued to $q^2>0$) reveals. Numerical results for the analytically continued infinite-momentum-frame form factors are found to agree very well with lattice data in the timelike momentum transfer region, and also, the experimental value for the slope of the $F_{B\to D}^{+}$ transition form factor at zero recoil is reproduced satisfactorily. Furthermore, these predictions satisfy heavy-quark-symmetry constraints, and their $q^2$ dependence is well approximated by a pole fit, reminiscent of a vector-meson-dominance-like decay mechanism. We discuss how such a decay mechanism can be accommodated within an extension of our constituent-quark model, by allowing for a nonvalence component in the meson wave functions, and we also address the question of wrong cluster properties inherent in the formulation of relativistic quantum mechanics employed in this article.

We report on ongoing work to determine the pion-cloud contribution to the electromagnetic $N\rightarrow\Delta$ transition form factors. The starting point is an $SU(6)$ spin-flavor symmetric constituent-quark model with instantaneous confinement that is augmented by dynamical pions which couple directly to the quarks. This system is treated in a relativistically invariant way within the framework of point-form quantum mechanics using a multichannel formulation. The first step is to determine the electromagnetic form factors of the bare particles that consist only of three quarks. These form factors are basic ingredients for calculating the pion-cloud contributions. Already without the pion cloud, electromagnetic nucleon and $N\rightarrow \Delta$ transition form factors compare reasonably well with the data. By inclusion of the pion-cloud contribution coming from the $\pi$-$N$ intermediate state the reproduction of the data is further improved.