arXiv subjects

Yaiza Canzani

Publications and source records attributed to Yaiza Canzani.

At least 19 recordsLinked to original sources

arXiv2026

Logarithmic improvements in the Weyl law and exponential bounds on the number of closed geodesics are predominant

Let $M$ be a smooth compact manifold of dimension $d$ without boundary. We introduce the concept of predominance for Riemannian metrics on $M$, a notion analogous to full Lebesgue measure which, in particular, implies density. We show that for a predominant metric, the number of closed geodesics of length smaller than $T$ has a stretched exponential upper bound in $T$. In addition, we study remainders in the Weyl law for predominant metrics. The Weyl law states that the number of Laplace-Beltrami eigenvalues smaller than $λ^2$ is asymptotic to $Cλ^d$ with an $O(λ^{d-1})$ error. We show that, for a predominant metric, the estimate on the error can by improved by a power of $\log λ$. After an application of recent results of the authors in the case of the Weyl law, these estimates follow from a study of the non-degeneracy properties of nearly closed orbits for predominant sets of metrics.

math.DS↗

arXiv2025

On non-local exchange and scattering operators in domain decomposition methods

We study non-local exchange and scattering operators arising in domain decomposition algorithms for solving elliptic problems on domains in $\mathbb{R}^2$. Motivated by recent formulations of the Optimized Schwarz Method introduced by Claeys, we rigorously analyze the behavior of a family of non-local exchange operators $Π_γ$, defined in terms of boundary integral operators associated to the fundamental solution for $-Δ+ γ^{-2}$, with $γ> 0$. Our first main result establishes precise estimates comparing $Π_γ$ to its local counterpart $Π_0$ as $γ\to 0$, providing a quantitative bridge between the classical and non-local formulations of the Optimized Schwarz Method. In addition, we investigate the corresponding scattering operators, proving norm estimates that relate them to their classical analogues through a detailed analysis of the associated Dirichlet-to-Neumann operators. Our results clarify the relationship between classical and non-local formulations of domain decomposition methods and yield new insights that are essential for the analysis of these algorithms, particularly in the presence of cross points and for domains with curvilinear polygonal boundaries.

math.NA↗

arXiv2024

Stability of spectral partitions with corners

A spectral minimal partition of a manifold is a decomposition into disjoint open sets that minimizes a spectral energy functional. While it is known that bipartite minimal partitions correspond to nodal partitions of Courant-sharp Laplacian eigenfunctions, the non-bipartite case is much more challenging. In this paper, we unify the bipartite and non-bipartite settings by defining a modified Laplacian operator and proving that the nodal partitions of its eigenfunctions are exactly the critical points of the spectral energy functional. Moreover, we prove that the Morse index of a critical point equals the nodal deficiency of the corresponding eigenfunction. Some striking consequences of our main result are: 1) in the bipartite case, every local minimum of the energy functional is in fact a global minimum; 2) in the non-bipartite case, every local minimum of the energy functional minimizes within a certain topological class of partitions. Our results are valid for partitions with non-smooth boundaries; this introduces considerable technical challenges, which are overcome using delicate approximation arguments in the Sobolev space $H^{1/2}$.

math.AP↗

arXiv2024

Asymptotics for the spectral function on Zoll manifolds

Let $(M,g)$ be a Zoll manifold, i.e., a smooth, compact, Riemannian manifold without boundary all of whose geodesics are closed with a minimal common period $T$. The positive definite Laplace-Beltrami operator has eigenvalues $\{λ_j^2\}_j$ which cluster around $ν^2_\ell$ for some sequence $ν_\ell\to \infty$. This article is concerned with the number of $λ_j$ in a window of fixed size $\mathrm{w}$ around $ν_\ell$, denoted by $\mathbf{N}(ν_\ell,\mathrm{w}):=\#\{j\,:\, λ_j\in[ν_\ell-\mathrm{w},ν_\ell+\mathrm{w}]\}.$ When the set of trajectories with period smaller than $T$ has zero measure, there is $c_{n}>0$, depending only on $n=\operatorname{dim} M$, such that $$ \mathbf{N}(ν_\ell,\mathrm{w}) =c_n\operatorname{vol}_g(M)ν_{\ell}^{n-1}+o(ν_{\ell}^{n-1}), $$ as $\ell \to \infty$. However, for a general Zoll manifold this may not be the case. We show that, nevertheless, there is $N>0$, independent of $\ell$, such that $$ \sum_{j=0}^{N-1}\mathbf{N}(ν_{\ell+j},\mathrm{w})= c_nN\operatorname{vol}_g(M)ν_{\ell}^{n-1}+o(ν_{\ell}^{n-1}), $$ as $\ell \to \infty$. In addition to asymptotics for the counting function, we study the kernel of the spectral projector for the Laplacian, $Π_{\ell,\mathrm{w}}(x,y)$ onto the spectrum in ${\bigcup_{j=0}^{N-1}[ν_{\ell+j}-\mathrm{w},ν_{\ell+j}+\mathrm{w}]}$. We show that for $x$ and $y$ in a shrinking neighborhood of a point with few loops of length smaller than $T$, $Π_{\ell,\mathrm{w}}(x,y)$ and its derivatives have the same asymptotics as those on the round sphere and flat torus.

math.AP↗

arXiv2024

Homology of spectral minimal partitions

A spectral minimal partition of a manifold is its decomposition into disjoint open sets that minimizes a spectral energy functional. It is known that bipartite spectral minimal partitions coincide with nodal partitions of Courant-sharp Laplacian eigenfunctions. However, almost all minimal partitions are non-bipartite. To study those, we define a modified Laplacian operator and prove that the nodal partitions of its Courant-sharp eigenfunctions are minimal within a certain topological class of partitions. This yields new results in the non-bipartite case and recovers the above known result in the bipartite case. Our approach is based on tools from algebraic topology, which we illustrate by a number of examples where the topological types of partitions are characterized by relative homology.

math.AP↗

arXiv2023

Uniform upper bounds on Courant sharp Neumann eigenvalues of chain domains

We obtain upper bounds on the number of nodal domains of Laplace eigenfunctions on chain domains with Neumann boundary conditions. The chain domains consist of a family of planar domains, with piecewise smooth boundary, that are joined by thin necks. Our work does not assume a lower bound on the width of the necks in the chain domain. As a consequence, we prove an upper bound on the number of Courant sharp eigenfunctions that is independent of the widths of the necks.

math.SP↗

arXiv2022

Stability of spectral partitions and the Dirichlet-to-Neumann map

The oscillation of a Laplacian eigenfunction gives a great deal of information about the manifold on which it is defined. This oscillation can be encoded in the nodal deficiency, an important geometric quantity that is notoriously hard to compute, or even estimate. Here we compare two recently obtained formulas for the nodal deficiency, one in terms of an energy functional on the space of equipartitions of the manifold, and the other in terms of a two-sided Dirichlet-to-Neumann map defined on the nodal set. We relate these two approaches by giving an explicit formula for the Hessian of the equipartition energy in terms of the Dirichlet-to-Neumann map. This allows us to compute Hessian eigenfunctions, and hence directions of steepest descent, for the equipartition energy in terms of the corresponding Dirichlet-to-Neumann eigenfunctions. Our results do not assume bipartiteness, and hence are relevant to the study of spectral minimal partitions.

math.AP↗

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.

math.AP↗

arXiv2022

Weyl remainders: an application of geodesic beams

We obtain new quantitative estimates on Weyl Law remainders under dynamical assumptions on the geodesic flow. On a smooth compact Riemannian manifold $(M,g)$ of dimension $n$, let $Π_λ$ denote the kernel of the spectral projector for the Laplacian, $\mathbb{1}_{[0,λ^2]}(-Δ_g)$. Assuming only that the set of near periodic geodesics over $W\subset M$ has small measure, we prove that as $λ\to \infty$ $$ \int_{W} Π_λ(x,x)dx=(2π)^{-n}\text{vol}_{\mathbb{R}^n}(B)\text{vol}_g(W)\,λ^n+O\Big(\frac{λ^{n-1}}{\log λ}\Big),$$ where $B$ is the unit ball. One consequence of this result is that the improved remainder holds on all product manifolds, in particular giving improved estimates for the eigenvalue counting function in the product setup. Our results also include logarithmic gains on asymptotics for the off-diagonal spectral projector $Π_λ(x,y)$ under the assumption that the set of geodesics that pass near both $x$ and $y$ has small measure, and quantitative improvements for Kuznecov sums under non-looping type assumptions. The key technique used in our study of the spectral projector is that of geodesic beams.

math.AP↗

arXiv2022

A local test for global extrema in the dispersion relation of a periodic graph

We consider a family of periodic tight-binding models (combinatorial graphs) that have the minimal number of links between copies of the fundamental domain. For this family we establish a local condition of second derivative type under which the critical points of the dispersion relation can be recognized as global maxima or minima. Under the additional assumption of time-reversal symmetry, we show that any local extremum of a dispersion band is in fact its global extremum if the dimension of the periodicity group is three or less, or (in any dimension) if the critical point in question is a symmetry point of the Floquet--Bloch family with respect to complex conjugation. We demonstrate that our results are nearly optimal with a number of examples.

math-ph↗

arXiv2021

Improvements for eigenfunction averages: An application of geodesic beams

Let $(M,g)$ be a smooth, compact Riemannian manifold and $\{ϕ_λ\}$ an $L^2$-normalized sequence of Laplace eigenfunctions, $-Δ_gϕ_λ=λ^2 ϕ_λ$. Given a smooth submanifold $H \subset M$ of codimension $k\geq 1$, we find conditions on the pair $(M,H)$, even when $H=\{x\}$, for which $$ \Big|\int_Hϕ_λdσ_H\Big|=O\Big(\frac{λ^{\frac{k-1}{2}}}{\sqrt{\log λ}}\Big)\qquad \text{or}\qquad |ϕ_λ(x)|=O\Big(\frac{λ^{\frac{n-1}{2}}}{\sqrt{\log λ}}\Big), $$ as $λ\to \infty$. These conditions require no global assumption on the manifold $M$ and instead relate to the structure of the set of recurrent directions in the unit normal bundle to $H$. Our results extend all previously known conditions guaranteeing improvements on averages, including those on sup-norms. For example, we show that if $(M,g)$ is a surface with Anosov geodesic flow, then there are logarithmically improved averages for any $H\subset M$. We also find weaker conditions than having no conjugate points which guarantee $\sqrt{\log λ}$ improvements for the $L^\infty$ norm of eigenfunctions. Our results are obtained using geodesic beam techniques, which yield a mechanism for obtaining general quantitative improvements for averages and sup-norms.

math.AP↗

arXiv2021

Quantitative bounds on Impedance-to-Impedance operators with applications to fast direct solvers for PDEs

We prove quantitative norm bounds for a family of operators involving impedance boundary conditions on convex, polygonal domains. A robust numerical construction of Helmholtz scattering solutions in variable media via the Dirichlet-to-Neumann operator involves a decomposition of the domain into a sequence of rectangles of varying scales and constructing impedance-to-impedance boundary operators on each subdomain. Our estimates in particular ensure the invertibility, with quantitative bounds in the frequency, of the merge operators required to reconstruct the original Dirichlet-to-Neumann operator in terms of these impedance-to-impedance operators of the sub-domains. A key step in our proof is to obtain Neumann and Dirichlet boundary trace estimates on solutions of the impedance problem, which are of independent interest. In addition to the variable media setting, we also construct bounds for similar merge operators in the obstacle scattering problem.

math.AP↗

arXiv2021

Growth of high $L^p$ norms for eigenfunctions: an application of geodesic beams

This work concerns $L^p$ norms of high energy Laplace eigenfunctions, $(-Δ_g-λ^2)ϕ_λ=0$, $\|ϕ_λ\|_{L^2}=1$. In 1988, Sogge gave optimal estimates on the growth of $\|ϕ_λ\|_{L^p}$ for a general compact Riemannian manifold. The goal of this article is to give general dynamical conditions guaranteeing quantitative improvements in $L^p$ estimates for $p>p_c$, where $p_c$ is the critical exponent. We also apply previous results of the authors to obtain quantitative improvements in concrete geometric settings including all product manifolds. These are the first results improving estimates for the $L^p$ growth of eigenfunctions that only require dynamical assumptions. In contrast with previous improvements, our assumptions are local in the sense that they depend only on the geodesics passing through a shrinking neighborhood of a given set in $M$. Moreover, the article gives a structure theorem for eigenfunctions which saturate the quantitatively improved $L^p$ bound. Modulo an error, the theorem describes these eigenfunctions as finite sums of quasimodes which, roughly, approximate zonal harmonics on the sphere scaled by $1/\sqrt{\log λ}$.

math.AP↗

arXiv2020

Eigenfunction concentration via geodesic beams

In this article we develop new techniques for studying concentration of Laplace eigenfunctions $ϕ_λ$ as their frequency, $λ$, grows. The method consists of controlling $ϕ_λ(x)$ by decomposing $ϕ_λ$ into a superposition of geodesic beams that run through the point $x$. Each beam is localized in phase-space on a tube centered around a geodesic whose radius shrinks slightly slower than $λ^{-\frac{1}{2}}$. We control $ϕ_λ(x)$ by the $L^2$-mass of $ϕ_λ$ on each geodesic tube and derive a purely dynamical statement through which $ϕ_λ(x)$ can be studied. In particular, we obtain estimates on $ϕ_λ(x)$ by decomposing the set of geodesic tubes into those that are non self-looping for time $T$ and those that are. This approach allows for quantitative improvements, in terms of $T$, on the available bounds for $L^\infty$ norms, $L^p$ norms, pointwise Weyl laws, and averages over submanifolds.

math.AP↗

arXiv2020

Local Universality for Zeros and Critical Points of Monochromatic Random Waves

This paper concerns the asymptotic behavior of zeros and critical points for monochromatic random waves $ϕ_λ$ of frequency $λ$ on a compact, smooth, Riemannian manifold $(M,g)$ as $λ\rightarrow \infty$. We prove that the measure of integration over the zero set of $ϕ_λ$ restricted to balls of radius $\approx λ^{-1}$ converges in distribution to the measure of integration over the zero set of a frequency $1$ random wave on $\mathbb R^n$, where $n$ is the dimension of $M$. We also prove convergence of finite moments for the counting measure of the critical points of ϕλ, again restricted to balls of radius $\approx λ^{-1}$, to the corresponding moments for frequency $1$ random waves. We then patch together these local results to obtain new global variance estimates on the volume of the zero set and numbers of critical points of $ϕ_λ$ on all of $M.$ Our local results hold under conditions about the structure of geodesics on $M$ that are generic in the space of all metrics on $M$, while our global results hold whenever $(M,g)$ has no conjugate points (e.g is negatively curved).

math.PR↗

arXiv2019

Nodal line estimates for the second Dirichlet eigenfunction

We study the nodal curves of low energy Dirichlet eigenfunctions in generalized curvilinear quadrilaterals. The techniques can be seen as a generalization of the tools developed by Grieser-Jerison in a series of works on convex planar domains and rectangles with one curved edge and a large aspect ratio. Here, we study the structure of the nodal curve in greater detail, in that we find precise bounds on its curvature, with uniform estimates up to the two points where it meets the domain at right angles, and show that many of our results hold for relatively small aspect ratios of the side lengths. We also discuss applications of our results to Courant-sharp eigenfunctions and spectral partitioning.

math.AP↗

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 $h \to 0^+$.

math.AP↗

arXiv2017

On the growth of eigenfunction averages: microlocalization and geometry

Let $(M,g)$ be a smooth, compact Riemannian manifold and $\{ϕ_h\}$ an $L^2$-normalized sequence of Laplace eigenfunctions, $-h^2Δ_gϕ_h=ϕ_h$. Given a smooth submanifold $H \subset M$ of codimension $k\geq 1$, we find conditions on the pair $(\{ϕ_h\},H)$ for which $$ \Big|\int_Hϕ_hdσ_H\Big|=o(h^{\frac{1-k}{2}}),\qquad h\to 0^+. $$ One such condition is that the set of conormal directions to $H$ that are recurrent has measure $0$. In particular, we show that the upper bound holds for any $H$ if $(M,g)$ is surface with Anosov geodesic flow or a manifold of constant negative curvature. The results are obtained by characterizing the behavior of the defect measures of eigenfunctions with maximal averages.

math.AP↗