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

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

Explaining the evolution of pension structure and job tenure

Average and expected job tenure of workers has fallen significantly over the last two decades. Workers have also experienced a major shift in pension coverage. Traditional defined benefit pensions, designed to reward long tenure, have become steadily less common, while defined contribution pensions, which are largely portable, have spread. The link between job tenure and pension trends has not been closely examined, but it offers insights about both phenomena. This paper uses a contract-theoretic matching model with moral hazard to explain changes in both pension structure and job tenure; we discuss how a richer model with job-specific human capital subject to technology shocks would yield similar results. In our model, a decline in the value of existing jobs relative to new jobs reduces expected match duration and thus the appeal of DB pensions. We argue that these trends are linked to changes in the nature of new technologies. This explanation is consistent with observed trends in technological change, tenure, and pension structure. Our results suggest an additional consequence of technological progress that has not been closely studied.Pensions ; Retirement ; Defined benefit pension plans ; Defined contribution pension plans

A Convergent Iterative Solution of the Quantum Double-well Potential

We present a new convergent iterative solution for the two lowest quantum wave functions $\psi_{ev}$ and $\psi_{od}$ of the Hamiltonian with a quartic double well potential $V$ in one dimension. By starting from a trial function, which is by itself the exact lowest even or odd eigenstate of a different Hamiltonian with a modified potential $V+\delta V$, we construct the Green's function for the modified potential. The true wave functions, $\psi_{ev}$ or $\psi_{od}$, then satisfies a linear inhomogeneous integral equation, in which the inhomogeneous term is the trial function, and the kernel is the product of the Green's function times the sum of $\delta V$, the potential difference, and the corresponding energy shift. By iterating this equation we obtain successive approximations to the true wave function; furthermore, the approximate energy shift is also adjusted at each iteration so that the approximate wave function is well behaved everywhere. We are able to prove that this iterative procedure converges for both the energy and the wave function at all $x$.Comment: 76 pages, Latex, no figure, 1 tabl