arXiv subjects

Madelyne M. Brown

Publications and source records attributed to Madelyne M. Brown.

3 recordsLinked to original sources

arXiv2025

$L^p$ estimates for joint quasimodes of two pseudodifferential operators whose characteristic sets have $k$-th order contact

On a smooth, compact, $n$-dimensional Riemannian manifold, we consider functions $u_h$ that are joint quasimodes of two semiclassical pseudodifferential operators $p_1(x,hD)$ and $p_2(x,hD)$. We develop $L^p$ estimates for $u_h$ when the characteristic sets of $p_1$ and $p_2$ meet with $k$-th order contact. This paper is the natural extension of the two-dimensional results from arXiv:1909.12559 to $n$ dimensions.

math.AP↗

arXiv2022

On the growth of generalized Fourier coefficients of restricted eigenfunctions

Let $(M,g)$ be a smooth, compact, Riemannian manifold and $\{ϕ_h\}$ 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 $ϕ_h|_H$ by testing them against a sequence of functions $\{ψ_h\}$ on $H$ whose wavefront set avoids $S^*H$. That is, we study what we call the generalized Fourier coefficients: $\langle ϕ_h,ψ_h\rangle_{L^2(H)}$. We give an explicit bound on these coefficients depending on how the defect measures for the two collections of functions $ϕ_h$ and $ψ_h$ relate. This allows us to get a little$-o$ improvement whenever the collection of recurrent directions over the wavefront set of $ψ_h$ is small. To obtain our estimates, we utilize geodesic beam techniques.

math.AP↗

arXiv2017

Spectrum of the Kohn Laplacian on the Rossi sphere

We study the spectrum of the Kohn Laplacian $\square_b^t$ on the Rossi example $(\mathbb{S}^3, \mathcal{L}_t)$. In particular we show that $0$ is in the essential spectrum of $\square_b^t$, which yields another proof of the global non-embeddability of the Rossi example.

math.CV↗