arXiv subjects

John A. Toth

Publications and source records attributed to John A. Toth.

At least 19 recordsLinked to original sources

arXiv2026

Non-concentration estimates for Laplace eigenfunctions on compact $C^{\infty}$ manifolds with boundary

Let $Ω$ be an $n$-dimensional compact Riemannian manifold $(n \geq 3)$ with $C^\infty$ boundary, and consider $L^2$-normalized eigenfunctions $ - Δϕ_λ = λ^2 ϕ_λ$ with Dirichlet or Neumann boundary conditions . In this note, we extend well-known interior nonconcentration bounds up to the boundary. Specifically, in Theorem \ref{thm1}, using purely stationary local methods, we prove that for such $Ω$ it follows that for {\em any} $x_0 \in \overlineΩ$ (including boundary points) and for all $μ\geq C_Ω λ^{-1}$ with sufficiently large constant $C_Ω >0,$ \begin{equation} \label{nonconbdy} \| ϕ_λ\|_{B(x_0,μ)\cap Ω}^2 = O(μ). \end{equation} In Theorem \ref{thm2} we extend a result of Sogge \cite{So} to manifolds with smooth boundary and show that \begin{equation} \label{SUPBD} \| ϕ_λ\|_{L^\infty(Ω)} \leq C λ^{\frac{n}{2}} \cdot \Big( \sup_{x \in Ω} \| ϕ_λ \|_{L^2( B(x,λ^{-1}) \cap Ω)} \Big). \end{equation} The sharp sup bounds $\| ϕ_λ \|_{L^\infty(Ω)} = O(λ^{\frac{n-1}{2}})$ for Dirichlet or Neumann eigenfunctions proved by Grieser in \cite{Gr} are then an immediate consequence of Theorems \ref{thm1} and \ref{thm2}.

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arXiv2025

$L^2$ restriction bounds for analytic continuations of quantum ergodic Laplace eigenfunctions

We prove a quantum ergodic restriction (QER) theorem for real hypersurfaces $Σ\subset X,$ where $X$ is the Grauert tube associated with a real-analytic, compact Riemannian manifold. As an application, we obtain $h$ independent upper and lower bounds for the $L^2$ - restrictions of the FBI transform of Laplace eigenfunctions restricted to $Σ$ satisfying certain generic geometric conditions.

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arXiv2023

Small-scale mass estimates for Neumann eigenfunctions: piecewise smooth planar domains

Let $Ω$ be a piecewise-smooth, bounded convex domain in $\R^2$ and consider $L^2$-normalized Neumann eigenfunctions $ϕ_λ$ with eigenvalue $λ^2$. Our main result is a small-scale {\em non-concentration} estimate: We prove that for {\em any} $x_0 \in \overlineΩ,$ (including boundary and corner points) and any $δ\in [0,1),$ $$ \| ϕ_λ\|_{B(x_0,λ^{-δ})\cap Ω} = O(λ^{-δ/2}).$$ The proof is a stationary vector field argument combined with a small scale induction argument.

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arXiv2022

Lower bounds for eigenfunction restrictions in lacunary regions

Let $(M,g)$ be a compact, smooth Riemannian manifold and $\{u_h\}$ be a sequence of $L^2$-normalized Laplace eigenfunctions that has a localized defect measure $μ$ in the sense that $ M \setminus \text{supp}(π_* μ) \neq \emptyset$ where $π:T^*M \to M$ is the canonical projection. Using Carleman estimates we prove that for any real-smooth closed hypersurface $H \subset (M\setminus \text{supp} (π_* μ))$ sufficiently close to $ \text{supp}(π_* μ),$ and for all $δ>0,$ $$ \int_{H} |u_h|^2 dσ\geq C_δ\, e^{- [\, d(H, \text{supp}(π_* μ)) + \,δ] /h} $$ as $h \to 0^+$. We also show that the result holds for eigenfunctions of Schrödinger operators and give applications to eigenfunctions on warped products and joint eigenfunctions of quantum completely integrable (QCI) systems.

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arXiv2021

Lower bounds for Steklov eigenfunctions

Let $(Ω,g)$ be a compact, analytic Riemannian manifold with analytic boundary $\partial Ω= M.$ We give $L^2$-lower bounds for Steklov eigenfunctions and their restrictions to interior hypersurfaces $H \subset Ω^{\circ}$ in a geometrically defined neighborhood of $M$. Our results are optimal in the entire geometric neighborhood and complement the results on eigenfunction upper bounds in the author's previous work.

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arXiv2020

Non-concentration and restriction bounds for Neumann eigenfunctions of piecewise $C^{\infty}$ bounded planar domains

Let $(Ω,g)$ be a piecewise-smooth, bounded convex domain in $\R^2$ and consider $L^2$-normalized Neumann eigenfunctions $ϕ_λ$ with eigenvalue $λ^2$ and $u_λ:= ϕ_λ |_{\partial Ω}$ the associated Dirichlet data (ie. boundary restriction of $ϕ_λ$). Our first main result (Theorem \ref{T:non-con}) is a small-scale {\em non-concentration} estimate: We prove that for {\em any} $x_0 \in \overlineΩ,$ (including boundary corner points) and any $δ\in [0,1),$ $$ \| ϕ_h \|_{B(x_0,λ^{-δ})\cap Ω} = O(λ^{-δ/2}).$$ Our subsequent results involve applications of the nonconcentration estimate to upper bounds for $L^2$ restrictions of boundary eigenfunctions that are valid up to boundary corners. In particular, in Theorem \ref{dirichlet} we prove that for any {\em flat} boundary edge $Γ$ (possibly including corner points), the boundary restrictions $u_h:= ϕ_h |_{\partial Ω}$ satisfy the bounds $$ \|u_λ \|_{L^2(Γ)} = O_ε(λ^{1/4 + ε}),$$ for any $ε>0.$ The exponent $1/4$ is sharp and the result improves on the $O(λ^{1/3})$ universal $L^2$-restriction bound for Neumann eigenfunctions due to Tataru \cite{Ta}. The $O(λ^{1/4})$ -bound is also an extension to the boundary (including corner points) of well-known interior $L^2$ restriction bounds of Burq-Gerard-Tzvetkov \cite{BGT} along totally-geodesic hypersurfaces.

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arXiv2019

Reverse Agmon estimates and nodal intersection bounds in forbidden regions

Let $(M,g)$ be a compact, Riemannian manifold and $V \in C^{\infty}(M; \mathbb{R})$. Given a regular energy level $E > \min V$, we consider $L^2$-normalized eigenfunctions, $u_h,$ of the Schrodinger operator $P(h) = - h^2 Δ_g + V - E(h)$ with $P(h) u_h = 0$ and $E(h) = E + o(1)$ as $h \to 0^+.$ The well-known Agmon-Lithner estimates \cite{Hel} are exponential decay estimates (ie. upper bounds) for eigenfunctions in the forbidden region $\{ V>E \}.$ The decay rate is given in terms of the Agmon distance function $d_E$ associated with the degenerate Agmon metric $(V-E)_+ \, g$ with support in the forbidden region. The point of this note is to prove a partial converse to the Agmon estimates (ie. exponential {\em lower} bounds for the eigenfunctions) in terms of Agmon distance in the forbidden region under a control assumption on eigenfunction mass in the allowable region $\{ V< E \}$ arbitrarily close to the caustic $ \{ V = E \}.$ We then give some applications to hypersurface restriction bounds for eigenfunctions in the forbidden region along with corresponding nodal intersection estimates.

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arXiv2018

Pointwise bounds for joint eigenfunctions of quantum completely integrable systems

Let $(M,g)$ be a compact Riemannian manifold and $P_1:=-h^2Δ_g+V(x)-E_1$ so that $dp_1\neq 0$ on $p_1=0$. We assume that $P_1$ is quantum completely integrable in the sense that there exist functionally independent pseuodifferential operators $P_2,\dots P_n$ with $[P_i,P_j]=0$, $i,j=1,\dots ,n$. We study the pointwise bounds for the joint eigenfunctions, $u_h$ of the system $\{P_i\}_{i=1}^n$ with $P_1u_h=E_1u_h+o(1)$. We first give polynomial improvements over the standard Hörmander bounds for typical points in $M$. In two and three dimensions, these estimates agree with the Hardy exponent $h^{-\frac{1-n}{4}}$ and in higher dimensions we obtain a gain of $h^{\frac{1}{2}}$ over the Hörmander bound. In our second main result, under a real-analyticity assumption on the QCI system, we give exponential decay estimates for joint eigenfunctions at points outside the projection of invariant Lagrangian tori; that is at points $x\in M$ in the "microlocally forbidden" region $p_1^{-1}(E_1)\cap \dots \cap p_n^{-1}(E_n)\cap T^*_xM=\emptyset.$ These bounds are sharp locally near the projection of the invariant tori.

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arXiv2018

Pointwise Bounds for Steklov Eigenfunctions

Let $(Ω,g)$ be a compact, real-analytic Riemannian manifold with real-analytic boundary $\partial Ω.$ The harmonic extensions of the boundary Dirchlet-to-Neumann eigenfunctions are called Steklov eigenfunctions. We show that the Steklov eigenfuntions decay exponentially into the interior in terms of the Dirichlet-to-Neumann eigenvalues and give a sharp rate of decay to first order at the boundary. The proof uses the Poisson representation for the Steklov eigenfunctions combined with sharp $h$-microlocal concentration estimates for the boundary Dirichlet-to-Neumann eigenfunctions near the cosphere bundle $S^*\partial Ω.$ These estimates follow from sharp estimates on the concentration of the FBI transforms of solutions to analytic pseudodifferential equations $Pu=0$ near the characteristic set $\{σ(P)=0\}$.

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arXiv2018

Averages of eigenfunctions over hypersurfaces

Let $(M,g)$ be a compact, smooth, Riemannian manifold and $\{ ϕ_h \}$ an $L^2$-normalized sequence of Laplace eigenfunctions with defect measure $μ$. Let $H$ be a smooth hypersurface. Our main result says that when $μ$ is $\textit{not}$ concentrated conormally to $H$, the eigenfunction restrictions to $H$ and the restrictions of their normal derivatives to $H$ have integrals converging to 0 as $h \to 0^+$.

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arXiv2018

Eigenfunction scarring and improvements in $L^{\infty}$ bounds

We study the relationship between $L^\infty$ growth of eigenfunctions and their $L^2$ concentration as measured by defect measures. In particular, we show that scarring in the sense of concentration of defect measure on certain submanifolds is incompatible with maximal $L^\infty$ growth. In addition, we show that a defect measure which is too diffuse, such as the Liouville measure, is also incompatible with maximal eigenfunction growth.

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arXiv2017

Nodal intersections and Geometric Control

This article contains a generalization of the authors' results on numbers of nodal points of eigenfunctions on "good curves" in analytic plane domains (arXiv:0710.0101). The term `good' means that the $L^2$ norms of restrictions of eigenfunctions of eigenvalue $λ^2$ to the curve are bounded below by $e^{- C λ}$. In this article, the result is generalized to all real analytic Riemannian manifolds $(M, g)$ of any dimension $m$ without boundary. Moreover, a similar lower bound is given for the Hausdorff $m-2$ measure of the intersection of the nodal set with a good real analytic hypersurface. Most of the article is devoted to giving a dynamical or geometric control condition for `goodness' of a hypersurface. The conditions are that the hypersurface $H$ be asymmetric with respect to geodesics and that the flowout of the unit vectors with footpoint on $H$ have full measure in $S^*M. $ This gives a partial answer to a question of Bourgain-Rudnick of characterizing hypersurfaces $H$ on which a sequence of eigenfunctions vanishes. We show that under our conditions, a positive density sequence cannot vanish on $H$ or even have smaller $L^2$ norms than $e^{- C λ}$

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arXiv2017

Nodal length of Steklov eigenfunctions on real-analytic Riemannian surfaces

We prove sharp upper and lower bounds for the nodal length of Steklov eigenfunctions on real-analytic Riemannian surfaces with boundary. The argument involves frequency function methods for harmonic functions in the interior of the surface as well as the construction of exponentially accurate approximations for the Steklov eigenfunctions near the boundary.

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arXiv2014

Smooth billiards with a large Weyl remainder

The celebrated Hardy-Landau lower bound for the error term in the Gauss's circle problem can be viewed as an estimate from below for the remainder in Weyl's law on a square, with either Dirichlet or Neumann boundary conditions. We prove an analogous estimate for smooth star-shaped planar domains admitting an appropriate one-parameter family of periodic billiard trajectories. Examples include ellipses and smooth domains of constant width. Our results confirm a prediction of P. Sarnak who proved a similar statement for surfaces without boundary. We also obtain lower bounds on the error term in higher dimensions. In this case, the main contribution to the Weyl remainder typically comes from the "big" singularity at zero of the wave trace. However, for certain domains, such as the Euclidean ball, the dimension of the family of periodic trajectories is large enough to dominate the contribution of the singularity at zero.

math.AP↗

arXiv2014

Intersection bounds for nodal sets of planar Neumann eigenfunctions with interior analytic curves

Let $ Ω\subset R^2$ be a bounded piecewise smooth domain and $ϕ_λ$ be a Neumann (or Dirichlet) eigenfunction with eigenvalue $λ^2$ and nodal set ${ N}_{ϕ_λ} = {x \in Ω; ϕ_λ(x) = 0}.$ Let $H \subset Ω$ be an interior $C^ω$ curve. Consider the intersection number $$ n(λ,H):= \# (H \cap N_{ϕ_λ} ).$$ We first prove that for general piecewise-analytic domains, and under an appropriate "goodness" condition on $H$, $$ n(λ,H) = {\mathcal O}_H(λ) (*)$$ as $λ\rightarrow \infty.$ We then prove that the bound in $(*)$ is satisfied in the case of quantum ergodic (QE) sequences of interior eigenfunctions, provided $Ω$ is convex and $H$ has strictly positive geodesic curvature.

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arXiv2013

Exterior mass estimates and $L^2$ restriction bounds for Neumann data along hypersurfaces

We study the problem of estimating the $L^2$ norm of Laplace eigenfunctions on a compact Riemannian manifold $M$ when restricted to a hypersurface $H$. We prove mass estimates for the restrictions of eigenfunctions $ϕ_h$, $(h^2 Δ- 1)ϕ_h = 0$, to $H$ in the region exterior to the coball bundle of $H$, on $h^δ$-scales ($0\leq δ< 2/3$). We use this estimate to obtain an $O(1)$ $L^2$-restriction bound for the Neumann data along $H.$ The estimate also applies to eigenfunctions of semiclassical Schrödinger operators.

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arXiv2012

Averaged Pointwise Bounds for Deformations of Schrodinger Eigenfunctions

Let (M,g) be a n-dimensional compact Riemannian manifold. We consider the magnetic deformations of semiclassical Schrodinger operators on M for a family of magnetic potentials that depends smoothly on $k$ parameters $u$, for $k \geq n$, and satisfies a generic admissibility condition. Define the deformed Schrodinger eigenfunctions to be the $u$-parametrized semiclassical family of functions on M that is equal to the unitary magnetic Schrodinger propagator applied to the Schrodinger eigenfunctions. The main result of this article states that the $L^2$ norms in $u$ of the deformed Schrodinger eigenfunctions are bounded above and below by constants, uniformly on $M$ and in $\hbar$. In particular, the result shows that this non-random perturbation "kills" the blow-up of eigenfunctions. We give, as applications, an eigenfunction restriction bound and a quantum ergodicity result.

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arXiv2009

About the blowup of quasimodes on Riemannian manifolds

On any compact Riemannian manifold $(M, g)$ of dimension $n$, the $L^2$-normalized eigenfunctions ${ϕ_λ}$ satisfy $||ϕ_λ||_{\infty} \leq C λ^{\frac{n-1}{2}}$ where $-Δϕ_λ = λ^2 ϕ_λ.$ The bound is sharp in the class of all $(M, g)$ since it is obtained by zonal spherical harmonics on the standard $n$-sphere $S^n$. But of course, it is not sharp for many Riemannian manifolds, e.g. flat tori $\R^n/Γ$. We say that $S^n$, but not $\R^n/Γ$, is a Riemannian manifold with maximal eigenfunction growth. The problem which motivates this paper is to determine the $(M, g)$ with maximal eigenfunction growth. In an earlier work, two of us showed that such an $(M, g)$ must have a point $x$ where the set ${\mathcal L}_x$ of geodesic loops at $x$ has positive measure in $S^*_x M$. We strengthen this result here by showing that such a manifold must have a point where the set ${\mathcal R}_x$ of recurrent directions for the geodesic flow through x satisfies $|{\mathcal R}_x|>0$. We also show that if there are no such points, $L^2$-normalized quasimodes have sup-norms that are $o(λ^{n-1)/2})$, and, in the other extreme, we show that if there is a point blow-down $x$ at which the first return map for the flow is the identity, then there is a sequence of quasi-modes with $L^\infty$-norms that are $Ω(λ^{(n-1)/2})$.

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