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We study the renormalized Nelson model for a scalar matter particle in a continuous confining potential interacting with a possibly massless quantized radiation field. When the radiation field is massless we impose a mild infrared…

We study the spectral problems associated with the finite-difference operators $H_N = 2 \cosh(p) + V_N(x)$, where $V_N(x)$ is an arbitrary polynomial potential of degree $N$. These systems can be regarded as a solvable deformation of the…

We consider Gaussian random eigenfunctions (Hermite functions) of fixed energy level of the isotropic semi-classical Harmonic Oscillator on ${\bf R}^n$. We calculate the expected density of zeros of a random eigenfunction in the…

We analyze a general class of self-adjoint difference operators $H_\varepsilon = T_\varepsilon + V_\varepsilon$ on $\ell^2(\varepsilon\mathbb{Z}^d)$, where $V_\varepsilon$ is a one-well potential and $\varepsilon$ is a small parameter. We…

This article addresses the microlocalization of eigenfunctions for the semiclassical Schr\"odinger operator $-h^2\Delta+V$ on closed Riemann surfaces with real bounded potentials. Our primary aim is to establish quantitative bounds on the…

A local time decay estimate of fractional Schr\"odinger operators with slowly decaying positive potentials are studied. It is shown that its resolvent is smooth near zero and the time propagator has fast local time decay which is very…

We derive explicit bounds for the remainder term in the local Weyl law for locally hyperbolic manifolds, we also give the estimates of the derivative of this remainder. We use these to obtain explicit bounds for the C^k-norms of the…

Let $(\Omega,g)$ be a piecewise-smooth, bounded convex domain in $\R^2$ and consider $L^2$-normalized Neumann eigenfunctions $\phi_{\lambda}$ with eigenvalue $\lambda^2$ and $u_{\lambda}:= \phi_{\lambda} |_{\partial \Omega}$ the associated…

In this paper, we describe the leftmost eigenvalue of the non-selfadjoint operator $\mathcal{A}_h = -h^2\Delta+iV(x)$ with Dirichlet boundary conditions on a smooth bounded domain $\Omega\subset\mathbb{R}^n\,$, as $h\rightarrow0\,$. $V$ is…

We refine the $L^p$ restriction estimates for Laplace eigenfunctions on a Riemannian surface, originally established by Burq, G\'erard, and Tzvetkov. First, we establish estimates for the restriction of eigenfunctions to arbitrary Borel…

Let $(M,g)$ be a smooth, compact, Riemannian manifold and $\{\phi_h\}$ a sequence of $L^2$-normalized Laplace eigenfunctions on $M$. For a smooth submanifold $H\subset M$, we consider the growth of the restricted eigenfunctions $\phi_h|_H$…

The purpose of this paper is to give a simple proof of sharp $L^\infty$ estimates for the eigenfunctions of the Dirichlet Laplacian on smooth compact Riemannian manifolds $(M,g)$ of dimension $n\ge 2$ with boundary $\partial M$ and then to…

Let $(S^2,g)$ be a convex surface of revolution and $H \subset S^2$ the unique rotationally invariant geodesic. Let $\varphi^\ell_m$ be the orthonormal basis of joint eigenfunctions of $\Delta_g$ and $\partial_\theta$, the generator of the…

We consider eigenfunctions of a semiclassical Schr{\"o}dinger operator on an interval, with a single-well type potential and Dirichlet boundary conditions. We give upper/lower bounds on the L^2 density of the eigenfunctions that are uniform…

This article is about two types of restrictions of eigenfunctions $\phi_j$ on a compact Riemannian manifold $(M,g)$: First, we restrict to a submanifold $H \subset M$, and expand the restriction $\gamma_H \phi_j$ in eigenfunctions $e_k$ of…

Let $(M,g)$ be a compact, smooth, Riemannian manifold and $\{ \phi_h \}$ an $L^2$-normalized sequence of Laplace eigenfunctions with defect measure $\mu$. Let $H$ be a smooth hypersurface. Our main result says that when $\mu$ is…

In this paper, we consider an inverse conductivity problem on a bounded domain $\Omega\subset\mathbb{R}^n$, $n\geq2$, also known as Electrical Impedance Tomography (EIT), for the case where unknown impenetrable obstacles are embedded into…

We investigate dispersive estimates for the Schr\"odinger operator $H=-\Delta +V$ with $V$ is a real-valued decaying potential when there are zero energy resonances and eigenvalues in four spatial dimensions. If there is a zero energy…

We prove several results of concentration for eigenfunctions in Toeplitz quantization. With mild assumptions on the regularity, we prove that eigenfunctions are $O(exp(-cN^{\delta}))$ away from the corresponding level set of the symbol,…

For a large class of complete, non-compact Riemannian manifolds, $(M,g)$, with boundary, we prove high energy resolvent estimates in the case where there is one trapped hyperbolic geodesic. As an application, we have the following local…