Related papers: Scattering phase asymptotics with fractal remainde…

We consider the asymptotics of the one-dimensional cubic nonlinear Schr\"odinger equation with an external potential $V$ that does not admit bound states. Assuming that $\jBra{x}^{2+}V(x) \in L^1$ and that $u$ is orthogonal to any…

Motivated by recent interest in the spectrum of the Laplacian of incomplete surfaces with isolated conical singularities, we consider more general incomplete m-dimensional manifolds with singularities on sets of codimension at least 2. With…

In this three-part monograph, we prove that steady, incompressible Navier-Stokes flows posed over the moving boundary, $y = 0$, can be decomposed into Euler and Prandtl flows in the inviscid limit globally in $[1,\infty) \times [0,\infty)$,…

Let $R_{\lambda,j}$ be the $j$-th Bessel--Riesz transform, where $n\geq 1$, $\lambda>0$, and $j=1,\ldots,n+1$. In this article, we establish a Weyl type asymptotic for $[M_f,R_{\lambda,j}]$, the commutator of $R_{\lambda,j}$ with…

Let (M, g) be a compact smooth Riemannian manifold. We obtain new off-diagonal estimates as {\lambda} tend to infinity for the remainder in the pointwise Weyl Law for the kernel of the spectral projector of the Laplacian onto functions with…

Let (M,g) a compact Riemannian $n$-dimensional manifold with umbilic boundary. It is well know that, under certain hypothesis, in the conformal class of g there are scalar-flat metrics that have the boundary of M as a constant mean…

In this paper, we study the quantitative recurrence properties in the case of $\mathbb{Z}$-extension of Axiom A flows on a Riemannian manifold. We study the asymptotic behavior of the first return time to a small neighborhood of the…

In this note we obtain semiclassical resolvent estimates for non-trapping long range perturbations of the Laplacian on asymptotically Euclidean manifolds. Our proof is based on a positive commutator argument which differs from Mourre-type…

Let $(M,g)$ be a smooth compact Riemannian manifold of dimension $n$ with smooth boundary $\partial M$. Suppose that $(M,g)$ admits a scalar-flat conformal metric. We prove that the supremum of the isoperimetric quotient over the…

Asymptotic results are derived for the number of random walks in alcoves of affine Weyl groups (which are certain regions in $n$-dimensional Euclidean space bounded by hyperplanes), thus solving problems posed by Grabiner [J. Combin. Theory…

We consider the resolvent on asymptotically Euclidean warped product manifolds in an appropriate 0-Gevrey class, with trapped sets consisting of only finitely many components. We prove that the high-frequency resolvent is either bounded by…

We numerically show fractal Weyl law behavior in an open Hamiltonian system that is described by a smooth potential and which supports numerous above-barrier resonances. This behavior holds even relatively far away from the classical limit.…

We consider a periodic array of resonators, formed from Euler-Bernoulli beams, attached to the surface of an elastic half-space. Earlier studies of such systems have concentrated on compressional resonators. In this paper we consider the…

In this paper, we consider the scattering theory for acoustic-type equations on non-compact manifolds with a single flat end. Our main purpose is to show an existence result of non-scattering energies. Precisely, we show a Weyl-type lower…

We consider a non-trapping $n$-dimensional Lorentzian manifold endowed with an end structure modeled on the radial compactification of Minkowski space. We find a full asymptotic expansion for tempered forward solutions of the wave equation…

Let $(M,g)$ be a smooth compact Riemannian manifold of dimension $n$ with smooth boundary $\partial M$, admitting a scalar-flat conformal metric. We prove that the supremum of the isoperimetric ratio over the scalar-flat conformal class is…

For a compact Riemannian manifold, Weyl's law describes the asymptotic behavior of the counting function of the eigenvalues of the associated Laplace operator. In this paper we discuss Weyl's law in the context of automorphic forms. The…

We prove several types of scaling results for Wigner distributions of spectral projections of the isotropic Harmonic oscillator on $\mathbb R^d$. In prior work, we studied Wigner distributions $W_{\hbar, E_N(\hbar)}(x, \xi)$ of individual…

Conrey, Farmer, Keating, Rubinstein, and Snaith, recently conjectured formulas for the full asymptotics of the moments of $L$-functions. In the case of the Riemann zeta function, their conjecture states that the $2k$-th absolute moment of…

We study the asymptotics of iterates of the transfer operator for non-uniformly hyperbolic $\alpha$-Farey maps. We provide a family of observables which are Riemann integrable, locally constant and of bounded variation, and for which the…