A Non-Concentration Estimate for Partially Rectangular Billiards

We consider quasimodes on planar domains with a partially rectangular boundary. We prove that for any ε₀ > 0, an 𝒪(λ^?ε₀ ) quasimode must have L ² mass in the “wings” (in phase space) bounded below by λ^?2?δ for any δ > 0. The proof uses the author's recent work on 0-Gevrey smooth domains to approximate quasimodes on C ^{1, 1} domains. There is an improvement for C ^{k, α} and C ^∞ domains.     

Eigenfunctions of Laplacian for phase estimation from wavefront gradient or curvature sensing

Modal cross coupling usually exists in wavefront estimation through Zernike polynomials. In order to cope with the problem, the eigenfunctions of Laplacian with Neumann boundary condition are proposed instead of Zernike polynomials to reconstruct phase from wavefront gradient or curvature sensing. It is proved theoretically that these modals can avoid modal cross coupling in both wavefront gradient sensing and curvature sensing. In wavefront gradient sensing, the coefficients of eigenfunctions of Laplacian can be obtained from the integral of the scalar product between the gradient of Laplacian's eigenfunctions and wavefront gradient signal. In wavefront curvature sensing, the coefficients of eigenfunctions of Laplacian can be calculated from the integral of the product of Laplacian's eigenfunctions and wavefront curvature signal. This approach is applicable on arbitrary apertures as long as eigenfunctions of Laplacian on apertures of arbitrary shape can be obtained.     

Stochastic Processes Associated with a Sum of the Lévy Laplacians

Abstract

In this paper, we give a stochastic expression of a semigroup generated by a sum of the Lévy Laplacians acting on a class of S-transforms of white noise distributions in terms of an infinite sequence of independent Brownian motions.     

Analytic eigendistributions and applications to homogeneous trees

An analytic distribution on is an element, ν, of the dual of the space of analytic functions on K. In particular, ν defines a linear functional on the polynomial ring . In this work, we study the converse problem: given a linear functional on , try to find a minimal set K such that ν extends to an analytic distribution on K. This study was motivated by the desire to generalize a result that allows the representation of functions on a homogeneous tree as integrals of z-harmonic functions oven a certain interval. A function f on a homogeneous tree T of degree q+1 is said to be z-harmonic, if μ₁f=zf, where μ₁ is the nearest neighbor averaging operator. It was proved in [Cohen, Colonna, Adv. Appl. Math. 20 (1998) 253–274] that if |f(v)|MC^|v| for constants M>0 and , then there exist z-harmonic functions k_z such that where I is the interval with endpoints . In the present paper, we study the case when the above exponential growth condition holds with , which necessitates replacing k_z(v) dz with an analytic distribution ν_v satisfying the z-harmonicity condition μ₁ν=zν. We show that to each function on the tree satisfying the above exponential growth condition there corresponds an eigendistribution on an elliptical region containing I as the interval between its foci.     

Spreading of quasimodes in the Bunimovich stadium

We consider Dirichlet eigenfunctions of the Bunimovich stadium , satisfying . Write where is the central rectangle and denotes the ``wings,' i.e., the two semicircular regions. It is a topic of current interest in quantum theory to know whether eigenfunctions can concentrate in as . We obtain a lower bound on the mass of in , assuming that itself is -normalized; in other words, the norm of is controlled by times the norm in . Moreover, if is an quasimode, the same result holds, while for an quasimode we prove that the norm of is controlled by times the norm in . We also show that the norm of may be controlled by the integral of along , where is a smooth factor on vanishing at . These results complement recent work of Burq-Zworski which shows that the norm of is controlled by the norm in any pair of strips contained in , but adjacent to .

     

Weighted eigenvalue problems for Hessian equations

In this paper we consider weighted eigenvalue problems for fully nonlinear elliptic equations involving Hessian operators. In particular we consider a singular weight, which behaves like a Hardy potential and we prove the existence of weak eigenfunctions.     

Hua operators and Poisson transform for bounded symmetric domains

Let Ω be a bounded symmetric domain of non-tube type in Cn with rank r and S its Shilov boundary. We consider the Poisson transform Psf(z) for a hyperfunction f on S defined by the Poisson kernel Ps(z,u)=s(h(z,z)^n/r/²|h(z,u)^n/r|), (z,u)×Ω×S, s∈C. For all s satisfying certain non-integral condition we find a necessary and sufficient condition for the functions in the image of the Poisson transform in terms of Hua operators. When Ω is the type I matrix domain in Mn,m(C) (n?m), we prove that an eigenvalue equation for the second order Mn,n-valued Hua operator characterizes the image.     

Bernstein–Nikolskii inequalities and Riesz interpolation formula on compact homogeneous manifolds

Bernstein–Nikolskii inequalities and Riesz interpolation formula are established for eigenfunctions of Laplace operators and polynomials on compact homogeneous manifolds.     

Eigenfunctions of the weighted Laplacian and a vanishing theorem on gradient steady Ricci soliton

The aim of this note has two folds. First, we show a gradient estimate of the higher eigenfunctions of the weighted Laplacian on smooth metric measure spaces. In the second part, we consider a gradient steady Ricci soliton and prove that there exists a positive constant c(n)$c(n)$ depending only on the dimension n   of the soliton such that there is no nontrivial harmonic 1-form (hence harmonic function) which is in L^p$L^p$ on such a soliton for any 2<p<c(n)$2<p<c(n)$.