Convergent Iterative Solutions of Schroedinger Equation for a Generalized Double Well Potential

We present an explicit convergent iterative solution for the lowest energy state of the Schroedinger equation with a generalized double well potential $V=\frac{g^2}{2}(x^2-1)^2(x^2+a)$. The condition for the convergence of the iteration procedure and the dependence of the shape of the groundstate wave function on the parameter $a$ are discussed.Comment: 23 pages, 7 figure

Iterative Solutions for Low Lying Excited States of a Class of Schroedinger Equation

The convergent iterative procedure for solving the groundstate Schroedinger equation is extended to derive the excitation energy and the wave function of the low-lying excited states. The method is applied to the one-dimensional quartic potential problem. The results show that the iterative solution converges rapidly when the coupling $g$ is not too small.Comment: 14 pages, 4 figure

Eisenstein Series on Covers of Odd Orthogonal Groups

We study the Whittaker coefficients of the minimal parabolic Eisenstein series on the $n$-fold cover of the split odd orthogonal group $SO_{2r+1}$. If the degree of the cover is odd, then Beineke, Brubaker and Frechette have conjectured that the $p$-power contributions to the Whittaker coefficients may be computed using the theory of crystal graphs of type C, by attaching to each path component a Gauss sum or a degenerate Gauss sum depending on the fine structure of the path. We establish their conjecture using a combination of automorphic and combinatorial-representation-theoretic methods. Surprisingly, we must make use of the type A theory, and the two different crystal graph descriptions of Brubaker, Bump and Friedberg available for type A based on different factorizations of the long word into simple reflections. We also establish a formula for the Whittaker coefficients in the even degree cover case, again based on crystal graphs of type C. As a further consequence, we establish a Lie-theoretic description of the coefficients for $n$ sufficiently large, thereby confirming a conjecture of Brubaker, Bump and Friedberg.Comment: 62 page

The Impact of Technological Change on Older Workers: Evidence from Data on Computer Use

New technologies like computers alter skill requirements. This paper explores two related effects of computers on older workers, who use computers less. The evolution of computer use in the Current Population Survey suggests that impending retirement reduces the incentive of older workers to acquire new skills. The Health and Retirement Study shows, further, that computer users retire later than non-users. This may arise because computer users choose to retire later and also because workers planning later retirement choose to acquire computer skills. Instrumental variables estimates suggest that computer use directly lowers the probability of retirement.

Whittaker Coefficients of Metaplectic Eisenstein Series

We study Whittaker coefficients for maximal parabolic Eisenstein series on metaplectic covers of split reductive groups. By the theory of Eisenstein series these coefficients have meromorphic continuation and functional equation. However they are not Eulerian and the standard methods to compute them in the reductive case do not apply to covers. For "cominuscule" maximal parabolics, we give an explicit description of the coefficients as Dirichlet series whose arithmetic content is expressed in an exponential sum. The exponential sum is then shown to satisfy a twisted multiplicativity, reducing its determination to prime power contributions. These, in turn, are connected to Lusztig data for canonical bases on the dual group using a result of Kamnitzer. The exponential sum at prime powers is then evaluated for generic Lusztig data. To handle the remaining degenerate cases, the evaluation of the exponential sum appears best expressed in terms of string data for canonical bases, as shown in a detailed example in $GL_4$. Thus we demonstrate that the arithmetic part of metaplectic Whittaker coefficients is intimately connected to the relations between these two expressions for canonical bases.Comment: 51 pages. To appear in GAF

Tokuyama-type formulas for type B

We obtain explicit formulas for the product of a deformed Weyl denominator with the character of an irreducible representation of the spin group $\rm{Spin}_{2r+1}({\mathbb C})$, which is an analogue of the formulas of Tokuyama for Schur polynomials and Hamel-King for characters of symplectic groups. To give these, we start with a symplectic group and obtain such characters using the Casselman-Shalika formula. We then analyze this using objects which are naturally attached to the metaplectic double cover of an odd orthogonal group, which also has dual group $\rm{Spin}_{2r+1}({\mathbb C})$.Comment: 34 pages. To appear in Israel J. of Mat

Jarlskog Invariant of the Neutrino Mapping Matrix

The Jarlskog Invariant $J_{\nu-map}$ of the neutrino mapping matrix is calculated based on a phenomenological model which relates the smallness of light lepton masses $m_e$ and $m_1$ (of $\nu_1$) with the smallness of $T$ violation. For small $T$ violating phase $\chi_l$ in the lepton sector, $J_{\nu-map}$ is proportional to $\chi_l$, but $m_e$ and $m_1$ are proportional to $\chi_l^2$. This leads to $J_{\nu-map} \cong {1/6}\sqrt{\frac{m_e}{m_\mu}}+O \bigg(\sqrt{\frac{m_em_\mu}{m_\tau^2}}\bigg)+O \bigg(\sqrt{\frac{m_1m_2}{m_3^2}}\bigg)$. Assuming $\sqrt{\frac{m_1m_2}{m_3^2}}<<\sqrt{\frac{m_e}{m_\mu}}$, we find $J_{\nu-map}\cong 1.16\times 10^{-2}$, consistent with the present experimental data.Comment: 19 page

Deviations of the Lepton Mapping Matrix from the Harrison-Perkins-Scott Form

We propose a simple set of hypotheses governing the deviations of the leptonic mapping matrix from the Harrison-Perkins-Scott (HPS) form. These deviations are supposed to arise entirely from a perturbation of the mass matrix in the charged lepton sector. The perturbing matrix is assumed to be purely imaginary (thus maximally $T$-violating) and to have a strength in energy scale no greater (but perhaps smaller) than the muon mass. As we shall show, it then follows that the absolute value of the mapping matrix elements pertaining to the tau lepton deviate by no more than $O((m_\mu/m_\tau)^2) \simeq 3.5 \times 10^{-3}$ from their HPS values. Assuming that $(m_\mu/m_\tau)^2$ can be neglected, we derive two simple constraints on the four parameters $\theta_{12}$, $\theta_{23}$, $\theta_{31}$, and $\delta$ of the mapping matrix. These constraints are independent of the details of the imaginary $T$-violating perturbation of the charged lepton mass matrix. We also show that the $e$ and $\mu$ parts of the mapping matrix have a definite form governed by two parameters $\alpha$ and $\beta$; any deviation of order $m_\mu/m_\tau$ can be accommodated by adjusting these two parameters.Comment: 31 pages, 2 figure