Zeros of orthogonal polynomials on the real line

Let $p_n(x)$ be orthogonal polynomials associated to a measure $d\mu$ of compact support in $R$. If $E\not\in supp(d\mu)$, we show there is a $\delta>0$ so that for all $n$, either $p_n$ or $p_{n+1}$ has no zeros in $(E-\delta, E+\delta)$. If $E$ is an isolated point of $supp(d\mu)$, we show there is a $\delta$ so that for all $n$, either $p_n$ or $p_{n+1}$ has at most one zero in $(E-\delta, E+\delta)$. We provide an example where the zeros of $p_n$ are dense in a gap of $supp(d\mu)$.Comment: (preliminary version

Weak convergence of CD kernels and applications

We prove a general result on equality of the weak limits of the zero counting measure, $d\nu_n$, of orthogonal polynomials (defined by a measure $d\mu$) and $\frac{1}{n} K_n(x,x) d\mu(x)$. By combining this with Mate--Nevai and Totik upper bounds on $n\lambda_n(x)$, we prove some general results on $\int_I \frac{1}{n} K_n(x,x) d\mu_s\to 0$ for the singular part of $d\mu$ and $\int_I |\rho_E(x) - \frac{w(x)}{n} K_n(x,x)| dx\to 0$, where $\rho_E$ is the density of the equilibrium measure and $w(x)$ the density of $d\mu$

Generalized Bounded Variation and Inserting point masses

Let $d\mu$ be a probability measure on the unit circle and $d\nu$ be the measure formed by adding a pure point to $d\mu$. We give a simple formula for the Verblunsky coefficients of $d\nu$ based on a result of Simon. Then we consider $d\mu_0$, a probability measure on the unit circle with $\ell^2$ Verblunsky coefficients $(\alpha_n (d\mu_0))_{n=0}^{\infty}$ of bounded variation. We insert $m$ pure points to $d\mu$, rescale, and form the probability measure $d\mu_m$. We use the formula above to prove that the Verblunsky coefficients of $d\mu_m$ are in the form \alpha_n(d\mu_0) + \sum_{j=1}^m \frac{\ol{z_j}^{n} c_j}{n} + E_n, where the $c_j$'s are constants of norm 1 independent of the weights of the pure points and independent of $n$; the error term $E_n$ is in the order of $o(1/n)$. Furthermore, we prove that $d\mu_m$ is of $(m+1)$-generalized bounded variation - a notion that we shall introduce in the paper. Then we use this fact to prove that \lim_{n \to \infty} \vp_n^*(z, d\mu_m) is continuous and is equal to $D(z, d\mu_m)^{-1}$ away from the pure points.Comment: To appear in Constructive Approximatio

Muon capture on deuteron and 3He

The muon capture reactions 2H(\mu^-,\nu_\mu)nn and 3He(\mu^-,\nu_\mu)3H are studied with conventional or chiral realistic potentials and consistent weak currents. The initial and final A=2 and 3 nuclear wave functions are obtained from the Argonne v18 or chiral N3LO two-nucleon potential, in combination with, respectively, the Urbana IX or chiral N2LO three-nucleon potential in the case of A=3. The weak current consists of polar- and axial-vector components. The former are related to the isovector piece of the electromagnetic current via the conserved-vector-current hypothesis. These and the axial currents are derived either in a meson-exchange or in a chiral effective field theory (chiEFT) framework. There is one parameter (either the N-to-\Delta axial coupling constant in the meson-exchange model, or the strength of a contact term in the chiEFT model) which is fixed by reproducing the Gamow-Teller matrix element in tritium beta-decay. The model dependence relative to the adopted interactions and currents (and cutoff sensitivity in the chiEFT currents) is weak, resulting in total rates of 392.0 +/- 2.3 Hz for A=2, and 1484 +/- 13 Hz for A=3, where the spread accounts for this model dependence.Comment: 15 pages, 1 figure, submitted to Phys. Rev.

Null tests from angular distributions in D→P1P2l+l−D \to P_1 P_2 l^+l^-, l=e,μl=e,\mu decays on and off peak

We systematically analyze the full angular distribution in $D \to P_1 P_2 l^+ l^-$ decays, where $P_{1,2}=\pi,K$, $l=e,\mu$. We identify several null tests of the standard model (SM). Notably, the angular coefficients $I_{5,6,7}$, driven by the leptons' axial-vector coupling $C_{10}^{(\prime)}$, vanish by means of a superior GIM-cancellation and are protected by parity invariance below the weak scale. CP-odd observables related to the angular coefficients $I_{5,6,8,9}$ allow to measure CP-asymmetries without $D$-tagging. The corresponding observables $A_{5,6,8,9}$ constitute null tests of the SM. Lepton universality in $|\Delta c| =|\Delta u|=1$ transitions can be tested by comparing $D \to P_1 P_2 \mu^+ \mu^-$ to $D \to P_1 P_2 e^+ e^-$ decays. Data for $P_1 P_2=\pi^+ \pi^-$ and $K^+ K^-$ on muon modes are available from LHCb and on electron modes from BESIII. Corresponding ratios of dimuon to dielectron branching fractions are at least about an order of magnitude away from probing the SM. In the future electron and muon measurements should be made available for the same cuts as corresponding ratios $R_{P_1 P_2}^D$ provide null tests of $e$-$\mu$-universality. We work out beyond-SM signals model-independently and in SM extensions with leptoquarks.Comment: 18 pages plus references and appendix, 7 figure