Related papers: Sign-balance of random Laplace eigenfunctions

A version of the Law of the Iterated Logarithm for smooth functions in the upper-half space is proved. As a consequence, we show that certain size conditions on the gradient and the gradient of the laplacian of a smooth function, lead to…

We study a large class of stochastic $p$-Laplace Allen-Cahn equations with singular potential. Under suitable assumptions on the (multiplicative-type) noise we first prove existence, uniqueness, and regularity of variational solutions.…

We construct harmonic functions in the quarter plane for discrete Laplace operators. In particular, the functions are conditioned to vanish on the boundary and the Laplacians admit coefficients associated with transition probabilities of…

In this paper we prove the existence of an optimal set for the minimization of the $k$-th variational eigenvalue of the $p$-Laplacian among $p$-quasi open sets of fixed measure included in a box of finite measure. An analogous existence…

We describe a new approach to understanding averages of high energy Laplace eigenfunctions, $u_h$, over submanifolds, $$ \Big|\int _H u_hd\sigma_H\Big| $$ where $H\subset M$ is a submanifold and $\sigma_H$ the induced by the Riemannian…

Let $S$ be a punctured Riemann surface with Euler characteristic $\chi(S)<0$. For any unitary representation $\rho: \pi_1(S) \to U(N)$, we introduce its renormalized energy and its harmonic representatives, which are equivariant harmonic…

We consider a weighted eigenvalue problem for the Dirichlet laplacian in a smooth bounded domain $\Omega\subset \mathbb{R}^N$, where the bang-bang weight equals a positive constant $\overline{m}$ on a ball $B\subset\Omega$ and a negative…

We give a sufficient condition for branched minimal immersions of spheres into ellipsoids to be embedded: we show that if the coordinate functions of the branched minimal immersion are first or second eigenfunctions with respect to a…

We show that in any harmonic space, the eigenvalue spectra of the Laplace operator on small geodesic spheres around a given point determine the norm $|\nabla R|$ of the covariant derivative of the Riemannian curvature tensor in that point.…

In this paper, we prove new pinching theorems for the first eigenvalue of the Laplacian on compact hypersurfaces of the Euclidean space. These pinching results are associated with the upper bound for the first eigenvalue in terms of higher…

We show that, with very high probability, the random graph Laplacian has simple spectrum. Our method provides a quantitatively effective estimate of the spectral gaps. Along the way, we establish results on affine no-gaps delocalization,…

We propose a numerical method to approximate the exact averaged boundary control of a family of wave equations depending on an unknown parameter sigma. More precisely the control, independent of sigma, that drives an initial data to a…

We study the high-energy eigenfunctions of the Laplacian on a compact Riemannian manifold with Anosov geodesic flow. The localization of a semiclassical measure associated with a sequence of eigenfunctions is characterized by the…

The convergence of the Rayleigh-Ritz method with nonlinear parameters optimized through minimization of the trace of the truncated matrix is demonstrated by a comparison with analytically known eigenstates of various quasi-solvable systems.…

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…

In this study, we show that the energy eigenvalues and the eigenfunctions of the Schrodinger equation for the power-law and the logarithmic potential can be easily obtained by using variation technique for special type wave functions. The…

Experimental data from a turbulent jet flow is analysed in terms of an additive, continuous stochastic process where the usual time variable is replaced by the scale. We show that the energy transfer through scales is well described by a…

Motivated by relativistic materials, we develop a numerical scheme to support existing or state new conjectures in the spectral optimisation of eigenvalues of the Dirac operator, subject to infinite-mass boundary conditions. We study the…

We prove pointwise bounds for $L^2$ eigenfunctions of the Laplace-Beltrami operator on locally symmetric spaces with $\mathbb{Q}$-rank one if the corresponding eigenvalues lie below the continuous part of the $L^2$ spectrum. Furthermore, we…

This paper is concerned with the Dirichlet eigenvalue problem for Laplace operator in a bounded domain with periodic perforation in the case of small volume. We obtain the optimal quantitative error estimates independent of the spectral…