相关论文: Quantum Leaks

We consider mass concentration properties of Laplace eigenfunctions $\varphi_\lambda$, that is, smooth functions satisfying the equation $-\Delta \varphi_\lambda = \lambda \varphi_\lambda$, on a smooth closed Riemannian manifold. Using a…

In this paper, we explore the geometric properties of unbounded extremal domains for the $p$-Laplacian operator in both Euclidean and hyperbolic spaces. Assuming that the nonlinearity grows at least as the nonlinearity of the eigenvalue…

We consider spaces of high-energy quasimodes for the Laplacian on a compact hyperbolic surface, and show that when the spaces are large enough, one can find quasimodes that exhibit strong localization phenomena. Namely, take any constant c,…

We prove that if $X$ is a finite area non-compact hyperbolic surface, then for any $\epsilon>0$, with probability tending to one as $n\to\infty$, a uniformly random degree $n$ Riemannian cover of $X$ has no eigenvalues of the Laplacian in…

The eigenfunctions of the Laplacian are a central object from the realms of analytic number theory to geometric analysis. We prove that H\"ormander $L^2$-$L^{\infty}$ estimates are equivalent to restriction estimates to small geodesic…

Is field space infinite? If not, it either loops back on itself or ends altogether. Periodic boundary conditions are of course familiar, but field space endpoints--which appear in real-world systems--are far less explored. In this paper we…

We discuss the most general effective Lagrangian obtained from the assumption that the degrees of freedom to be quantized, in a black hole, are on the horizon. The effective Lagrangian depends only on the induced metric and the extrinsic…

Consider the 3-dimensional Laplacian with a potential described by point scatterers placed on the integer lattice. We prove that for Floquet-Bloch modes with fixed quasi-momentum satisfying a certain Diophantine condition, there is a…

This paper contains a very simple and general proof that eigenfunctions of quantizations of classically ergodic systems become uniformly distributed in phase space. This ergodicity property of eigenfunctions f is shown to follow from a…

The asymptotic behavior of the first eigenvalues of magnetic Laplacian operators with large magnetic fields and Neumann realization in smooth three-dimensional domains is characterized by model problems inside the domain or on its boundary.…

On a closed hyperbolic surface, we investigate semiclassical defect measures associated with the magnetic Laplacian in the presence of a constant magnetic field. Depending on the energy level where the eigenfunctions concentrate, three…

We investigate the asymptotic behavior of eigenfunctions of the Laplacian on Riemannian manifolds. We show that Benjamini-Schramm convergence provides a unified language for the level and eigenvalue aspects of the theory. As a result, we…

Given any compact hyperbolic surface $M$, and a closed geodesic on $M$, we construct of a sequence of quasimodes on $M$ whose microlocal lifts concentrate positive mass on the geodesic. Thus, the Quantum Unique Ergodicity (QUE) property…

We consider a general d-dimensional quantum system of non-interacting particles, with suitable statistics, in a very large (formally infinite) container. We prove that, in equilibrium, the fluctuations in the density of particles in a…

For pure states of multi-dimensional quantum lattice systems, which in a convenient computational basis have amplitude and phase structure of sufficiently rapid decorrelation, we construct high fidelity approximations of relatively low…

We present several applications of mode matching methods in spectral and scattering problems. First, we consider the eigenvalue problem for the Dirichlet Laplacian in a finite cylindrical domain that is split into two subdomains by a…

We investigate the asymptotic spectral distribution of the twisted Laplacian associated with a real harmonic 1-form on a compact hyperbolic surface. In particular, we establish a sublinear lower bound on the number of eigenvalues in a…

This paper is concerned with the location of nodal sets of eigenfunctions of the Dirichlet Laplacian in thin tubular neighbourhoods of hypersurfaces of the Euclidean space of arbitrary dimension. In the limit when the radius of the…

We examine a specific category of eigenfunctions of the lattice Laplacian on $\{p,q\}$-tessellations of the Poincar\'e disk that bear resemblance to plane waves in the continuum case. Our investigation reveals that the lattice eigenmodes…

We prove a quantum ergodicity theorem for sequences of closed hyperbolic surfaces converging to the Poincar\'e disc in the Benjamini-Schramm sense. Assuming a uniform lower bound on the injectivity radius and a spectral gap, we establish…