Related papers: Eigenfunctions for quasi-laplacian

We develop a formalism to extend our previous work on the electromagnetic $\delta$-function plates to a spherical surface. The electric ($\lambda_e$) and magnetic ($\lambda_g$) couplings to the surface are through $\delta$-function…

Let $D^2$ be the open unit disc in the Euclidean plane and let $G:= Diff(D2; area)$ be the group of smooth compactly supported area-preserving diffeomorphisms of $D^2$. We investigate the properties of G endowed with the autonomous metric.…

This paper concerns the global in time existence of solutions for a semilinear heat equation \begin{equation} \tag{P} \label{eq:P} \begin{cases} \partial_t u = \Delta u + f(u), &x\in \mathbb{R}^N, \,\,\, t>0, \\[3pt] u(x,0) = u_0(x) \ge 0,…

Let $M$ be a compact connected manifold of dimension $n$ endowed with a conformal class $C$ of Riemannian metrics of volume one. For any integer $k\geq0$, we consider the conformal invariant $\lambda_k ^c (C)$ defined as the supremum of the…

Consider a semiclassical Hamiltonian \begin{equation*} H_{V, h} := h^{2} \Delta + V - E \end{equation*} where $h > 0$ is a semiclassical parameter, $\Delta$ is the positive Laplacian on $\mathbb{R}^{d}$, $V$ is a smooth, compactly supported…

We investigate Stein-Malliavin approximations for nonlinear functionals of geometric interest of Gaussian random eigenfunctions on the unit $d$ -dimensional sphere ${\mathbb{S}}^{d},$ $d\geq 2.$ All our results are established in the high…

Let $(M,g)$ be a complete manifold of nonpositive scalar curvature, let $\Omega\subset M$ be a suitable domain, and let $\lambda(\Omega)$ be the first Dirichlet eigenvalue of the Laplace-Beltrami operator on $\Omega$. We prove several…

By methods of stochastic analysis on Riemannian manifolds, we develop two approaches to determine an explicit constant $c(D)$ for an $n$-dimensional compact manifold $D$ with boundary such that $\frac{\lambda}{n}\,\|\phi\|_{\infty} \leq…

We consider the Nonlinear Schr\"odinger (NLS) equation and prove that the Gaussian measure with covariance $(1-\partial_x^2)^{-\alpha}$ on $L^2(\mathbf T)$ is quasi-invariant for the associated flow for $\alpha>1/2$. This is sharp and…

We study the (massless) Dirac operator on a 3-sphere equipped with Riemannian metric. For the standard metric the spectrum is known. In particular, the eigenvalues closest to zero are the two double eigenvalues +3/2 and -3/2. Our aim is to…

In the case of the sphere and the square, Antonie Stern (1925) claimed in her PhD thesis the existence of an infinite sequence of eigenvalues whose corresponding eigenspaces contain an eigenfunction with two nodal domains. These two…

Novel results for the self-consistent single-particle spectral function and self-energy are presented for non-degenerate one-component Coulomb systems at various densities and temperatures. The GW^0-method for the dynamical self-energy is…

We study $\ell^\infty$ norms of $\ell^2$-normalized eigenfunctions of quantum cat maps. For maps with short quantum periods (constructed by Bonechi and de Bi\`evre), we show that there exists a sequence of eigenfunctions $u$ with…

Let $\Delta$ be a 2-sphere endowed with an affine structure away from a finite set of points $P \subset \Delta$, and assume that the monodromy of the associated connection $\nabla$ on $\Delta \setminus P$ around any point from $P$ is…

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 purpose of this note is to study the connectedness at infinity of manifold by using the theory of $p$-harmonic functions. We show that if the first eigenvalue $\lambda_{1,p}$ for the $p$-Laplacian achievies its maximal value on a…

Let $S$ be a Damek-Ricci space equipped with the Laplace-Beltrami operator $\Delta$. In this paper we characterize all eigenfunctions of $\Delta $ through sphere, ball and shell averages as the radius (of sphere, ball or shell) tends to…

Let $g$ and $\tilde{g}$ be Riemannian metrics on a noncompact manifold $M$, which are conformally equivalent. We show that under a very mild \emph{first order} control on the conformal factor, the wave operators corresponding to the…

We develop a singular pseudodifferential calculus. The symbols that we consider do not satisfy the standard decay with respect to the frequency variables. We thus adopt a strategy based on the Calderon-Vaillancourt Theorem. The remainders…

We prove upper bounds on the $L^p$ norms of eigenfunctions of the discrete Laplacian on regular graphs. We then apply these ideas to study the $L^p$ norms of joint eigenfunctions of the Laplacian and an averaging operator over a finite…