Dispersive estimate for the Schroedinger equation with point interactions

We consider the Schroedinger operator in R^3 with N point interactions placed at Y=(y_1, ... ,y_N), y_j in R^3, of strength a=(a_1, ... ,a_N). Exploiting the spectral theorem and the rather explicit expression for the resolvent we prove a (weighted) dispersive estimate for the corresponding Schroedinger flow. In the special case N=1 the proof is directly obtained from the unitary group which is known in closed form.Comment: 12 page

Restoration of Isotropy in the Ising Model on the Sierpinski Gasket

We study the ferromagnetic Ising model on the Sierpinski gasket (SG), where the spin-spin interactions depends on the direction. Using the renormalization group method, we show that the ratios of the correlation lengths restore the isotropy of the lattice as the temperature approaches zero. This restoration is either partial or perfect, depending on the interactions. In case of partial restoration, we also evaluate the leading-order singular behavior of the correlation lengths.Comment: 17 pages, 10 figures. References added in v.2 and 3. Small improvements in v.4, 5. This version will appear in Prog. Theor. Phy

On the LpL^p boundedness of wave operators for two-dimensional Schr\"odinger operators with threshold obstructions

Let $H=-\Delta+V$ be a Schr\"odinger operator on $L^2(\mathbb R^2)$ with real-valued potential $V$, and let $H_0=-\Delta$. If $V$ has sufficient pointwise decay, the wave operators $W_{\pm}=s-\lim_{t\to \pm\infty} e^{itH}e^{-itH_0}$ are known to be bounded on $L^p(\mathbb R^2)$ for all $1< p< \infty$ if zero is not an eigenvalue or resonance. We show that if there is an s-wave resonance or an eigenvalue only at zero, then the wave operators are bounded on $L^p(\mathbb R^2)$ for $1 < p<\infty$. This result stands in contrast to results in higher dimensions, where the presence of zero energy obstructions is known to shrink the range of valid exponents $p$.Comment: Revised according to referee's comments. 22 pages, to appear in J. Funct. Ana

On nonparametric and semiparametric testing for multivariate linear time series

We formulate nonparametric and semiparametric hypothesis testing of multivariate stationary linear time series in a unified fashion and propose new test statistics based on estimators of the spectral density matrix. The limiting distributions of these test statistics under null hypotheses are always normal distributions, and they can be implemented easily for practical use. If null hypotheses are false, as the sample size goes to infinity, they diverge to infinity and consequently are consistent tests for any alternative. The approach can be applied to various null hypotheses such as the independence between the component series, the equality of the autocovariance functions or the autocorrelation functions of the component series, the separability of the covariance matrix function and the time reversibility. Furthermore, a null hypothesis with a nonlinear constraint like the conditional independence between the two series can be tested in the same way.Comment: Published in at http://dx.doi.org/10.1214/08-AOS610 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org