Related papers: Refined and microlocal Kakeya-Nikodym bounds for e…
We study two-dimensional eigenvalue ensembles close to certain types of singular points in the bulk of the droplet. We prove existence of a microscopic density which quickly approaches the classical equilibrium density, as the distance from…
We present explicit formulas for the Faddeev eigenfunctions and related generalized scattering data for multipoint potentials in two and three dimensions. For single point potentials in 3D such formulas were obtained in an old unpublished…
The purpose of this note is to extend to any space dimension the bilinear estimate for eigenfunctions of the Laplace operator on a compact manifold (without boundary) obtained in a previous work in dimension 2. We also give some related…
We study the nodal curves of low energy Dirichlet eigenfunctions in generalized curvilinear quadrilaterals. The techniques can be seen as a generalization of the tools developed by Grieser-Jerison in a series of works on convex planar…
Let $(M,g)$ be a smooth, compact Riemannian manifold and $\{\phi_h\}$ an $L^2$-normalized sequence of Laplace eigenfunctions, $-h^2\Delta_g\phi_h=\phi_h$. Given a smooth submanifold $H \subset M$ of codimension $k\geq 1$, we find conditions…
The class of differential-equation eigenvalue problems $-y''(x)+x^{2N+2}y(x)=x^N Ey(x)$ ($N=-1,0,1,2,3,...$) on the interval $-\infty<x<\infty$ can be solved in closed form for all the eigenvalues $E$ and the corresponding eigenfunctions…
We study the computation of local approximations of invariant manifolds of parabolic fixed points and parabolic periodic orbits of periodic vector fields. If the dimension of these manifolds is two or greater, in general, it is not possible…
We use density--functional theory to study the structure of two-dimensional defects inside a circular nematic nanocavity. The density, nematic order parameter, and director fields, as well as the defect core energy and core radius, are…
For an image pixel information can be converted to the moments of some basis $Q_k$, e.g. Fourier-Mellin, Zernike, monomials, etc. Given sufficient number of moments pixel information can be completely recovered, for insufficient number of…
We expand on counterxamples of Bourgain showing how Nikodym maximal estimates and oscillatory integral estimates can break down in the non-Euclidean case. Our examples show the role that the non-existence of totally geodesic submanifolds…
We consider the problem of the construction of the nanophotonic structures of arbitrary geometry with prescribed desired properties. We reformulate this problem as an optimization problem for the Tikhonov functional which is minimized on…
We consider a general energy functional for phase coexistence models, which comprises the case of Banach norms in the gradient term plus a double-well potential. We establish density estimates for $Q$-minima. Namely, the state parameters…
We use superspace methods to study an SYK-like model with $\mathcal N=2$ supersymmetry in one dimension, and an analog of this model in two dimensions. We find the four-point function as an expansion in the basis of eigenfunctions of the…
The article provides proofs for the regularity of Dirac eigenfunctions, subject to MIT boundary conditions employed on various types of open sets ranging from smooth ones to convex polygons in two dimensions, as well as on half-space and…
We study the decay rate for the energy of solutions of a damped wave equation in a situation where the Geometric Control Condition is violated. We assume that the set of undamped trajectories is a flat torus of positive codimension and that…
We consider error estimates in weak parametrised norms for stabilized finite element approximations of the two-dimensional Navier-Stokes' equations. These weak norms can be related to the norms of certain filtered quantities, where the…
Given an irrational rotation $T$ on $\M T$ we settle necessary and sufficient conditions on a step function $\phi$ and $t\in \M T$ for the existence of measurable solutions to the cohomogical equation $$\exp{(2i\pi\phi)}=\e{2i\pi t}f/f\rond…
There exist two major subclasses in the class of superquadratic functions, one comprises concave and decreasing functions, while the other consists of convex and monotone increasing functions. Leveraging this distinction, we introduce…
This paper presents a family of Fourier eigenfunctions indexed by the space dimension d. These eigenfunctions are radial and built upon some generalized exponential integral function. For d=1,2,3, they are integrable or square integrable…
We prove sharp L^2 boundary decay estimates for the eigenfunctions of certain second order elliptic operators acting in a bounded region, and of their first order space derivatives, using only the Hardy inequality. We then deduce bounds on…