arXiv subjects

Steve Zelditch

Publications and source records attributed to Steve Zelditch.

At least 37 records · Page 2Linked to original sources

arXiv2019

Boundedness of the number of nodal domains for eigenfunctions of generic Kaluza-Klein $3$-folds

This article concerns the number of nodal domains of eigenfunctions of the Laplacian on special Riemannian $3$-manifolds, namely nontrivial principal $S^1$ bundles $P \to X$ over Riemann surfaces equipped with certain $S^1$ invariant metrics, the Kaluza-Klein metrics. We prove for generic Kaluza-Klein metrics that any Laplacian eigenfunction has exactly two nodal domains unless it is invariant under the $S^1$ action. We also construct an explicit orthonormal eigenbasis on the flat $3$-torus $\mathbb{T}^3$ for which every non-constant eigenfunction belonging to the basis has two nodal domains.

math.SP↗

arXiv2019

Topology of the nodal set of random equivariant spherical harmonics on $S^3$

We show that real and imaginary parts of equivariant spherical harmonics on $S^3$ have almost surely a single nodal component. Moreover, if the degree of the spherical harmonic is $N$ and the equivariance degree is $m$, then the expected genus is proportional to $m \left(\frac{N^2 - m^2}{2} + N\right) $. Hence if $\frac{m}{N}= c $ for fixed $0 < c < 1$, the genus has order $N^3$.

math.DG↗

arXiv2019

Interface Asymptotics of Wigner-Weyl Distributions for the Harmonic Oscillator

We prove several types of scaling results for Wigner distributions of spectral projections of the isotropic Harmonic oscillator on $\mathbb R^d$. In prior work, we studied Wigner distributions $W_{\hbar, E_N(\hbar)}(x, ξ)$ of individual eigenspace projections. In this continuation, we study Weyl sums of such Wigner distributions as the eigenvalue $E_N(\hbar)$ ranges over spectral intervals $[E - δ(\hbar), E + δ(\hbar)]$ of various widths $δ(\hbar)$ and as $(x, ξ) \in T^*\mathbb R^d$ ranges over tubes of various widths around the classical energy surface $Σ_E \subset T^*\mathbb R^d$. The main results pertain to interface Airy scaling asymptotics around $Σ_E$, which divides phase space into an allowed and a forbidden region. The first result pertains to $δ(\hbar) = \hbar$ widths and generalizes our earlier results on Wigner distributions of individual eigenspace projections. Our second result pertains to $δ(\hbar) = \hbar^{2/3}$ spectral widths and Airy asymptotics of the Wigner distributions in $\hbar^{2/3}$-tubes around $Σ_E$. Our third result pertains to bulk spectral intervals of fixed width and the behavior of the Wigner distributions inside the energy surface, outside the energy surface and in a thin neighborhood of the energy surface.

math-ph↗

arXiv2019

Interface Asymptotics of Eigenspace Wigner distributions for the Harmonic Oscillator

Eigenspaces of the quantum isotropic Harmonic Oscillator $\hat{H}_{\hbar} : = - \frac{\hbar^2}{2} Δ+ \frac{||x||^2}{2}$ on $\mathbb{R}^d$ have extremally high multiplicites and the eigenspace projections $Π_{\hbar, E_N(\hbar)} $ have special asymptotic properties. This article gives a detailed study of their Wigner distributions $W_{\hbar, E_N(\hbar)}(x, ξ)$. Heuristically, if $E_N(\hbar) = E$, $W_{\hbar, E_N(\hbar)}(x, ξ)$ is the `quantization' of the energy surface $Σ_E$, and should be like the delta-function $δ_{Σ_E}$ on $Σ_E$; rigorously, $W_{\hbar, E_N(\hbar)}(x, ξ)$ tends in a weak* sense to $δ_{Σ_E}$. But its pointwise asymptotics and scaling asymptotics have more structure. The main results give Bessel asymptotics of $W_{\hbar, E_N(\hbar)}(x, ξ)$ in the interior $H(x, ξ) < E$ of $Σ_E$; interface Airy scaling asymptotics in tubes of radius $\hbar^{2/3}$ around $Σ_E$, with $(x, ξ)$ either in the interior or exterior of the energy ball; and exponential decay rates in the exterior of the energy surface.

math-ph↗

arXiv2018

Schrödinger Trace Invariants for Homogeneous Perturbations of the Harmonic Oscillator

Let $H = H_0 + P$ denote the harmonic oscillator on $\mathbb{R}^d$ perturbed by an isotropic pseudodifferential operator $P$ of order $1$ and let $U(t) = \operatorname{exp}(- it H)$. We prove a Gutzwiller-Duistermaat-Guillemin type trace formula for $\operatorname{Tr} U(t)$. The singularities occur at times $t \in 2 π\mathbb{Z}$ and the coefficients involve the dynamics of the Hamilton flow of the symbol $σ(P)$ on the space $\mathbb{CP}^{d-1}$ of harmonic oscillator orbits of energy $1$. This is a novel kind of sub-principal symbol effect on the trace. We generalize the averaging technique of Weinstein and Guillemin to this order of perturbation, and then present two completely different calculations of $\operatorname{Tr} U(t)$. The first proof directly constructs a parametrix of $U(t)$ in the isotropic calculus, following earlier work of Doll-Gannot-Wunsch. The second proof conjugates the trace to the Bargmann-Fock setting, the order $1$ of the perturbation coincides with the `central limit scaling' studied by Zelditch-Zhou for Toeplitz operators.

math.AP↗

arXiv2018

Log-scale equidistribution of nodal sets in Grauert tubes

Let $M_{τ_0}$ be the Grauert tube (of some fixed radius $τ_0$) of a compact, negatively curved, real analytic Riemannian manifold $M$ without boundary. Let $ϕ_λ$ be a Laplacian eigenfunction on $M$ of eigenvalues $-λ^2$ and let $ϕ_λ^\mathbb{C}$ be its holomorphic extension to $M_{τ_0}$. In this article, we prove that on $M_{τ_0} \setminus M$, there exists a dimensional constant $α> 0$ and a full density subsequence $ \{λ_{j_k}\}_{k=1}^{\infty}$ of the spectrum for which the masses of the complexified eigenfunctions $ϕ_{λ_{j_k}}^\mathbb{C}$ are asymptotically equidistributed at length scale $(\log λ_{j_k})^{-α}$. Moreover, the complex zeros of $ϕ_{λ_{j_k}}^\mathbb{C}$ also become equidistributed on this logarithmic length scale.

math.AP↗

arXiv2018

Log-scale equidistribution of zeros of quantum ergodic eigensections

Under suitable hypotheses, a symplectic map can be quantized as a sequence of unitary operators acting on the $N$th powers of a positive line bundle over a Kähler manifold. We show that if the symplectic map has polynomial decay of correlations, then there exists a density one subsequence of eigensections whose masses and zeros become equidistributed in balls of logarithmically shrinking radii of lengths $\lvert \log N \rvert^{-γ}$ for some constant $γ> 0$ independent of $N$.

math.SP↗

arXiv2018

Local and global analysis of nodal sets

This is a survey for the JDG 50th Anniversary conference of recent results on nodal sets of eigenfunctions of the Laplacian on a compact Riemannian manifold. In part the techniques are `local', i.e. only assuming eigenfunctions are defined on small balls, and in part the techniques are `global', i.e. exploiting dynamics of the geodesic flow. The local part begins with a review of doubling indices and freqeuency functions as local measures of fast or slow growth of eigenfunctions. The pattern of boxes with maximal doubling indices plays a central role in the results of Logunov-Malinnokova, giving upper and lower bounds for hypersurface measures of nodal sets in the setting of general $C^{\infty}$ metrics. The proofs of both their polynomial upper bound and sharp lower bound are sketched. The survey continues with a global proof of the sharp upper bound for real analytic metrics (originally proved by Donnelly-Fefferman with local arguments), using analytic continuation to Grauert tubes. Then it reviews results of Toth-Zelditch giving sharp upper bounds on Hausdorff measures of intersections of nodal sets with real analytic submanifolds in the real analytic setting. Last, it goes over lower bounds of Jung-Zelditch on numbers of nodal domains in the case of $C^{\infty}$ surfaces of non-positive curvature and concave boundary or on negatively curved `real' Riemann surfaces, which are based on ergodic properties of the geodesic flow and eigenfunctions, and on estimates of restrictions of eigenfunctions to hypersurfaces. The last section details recent results on restrictions.

math.AP↗

arXiv2018

Pointwise Weyl laws for Partial Bergman kernels

This is a partly expository article for the volume "Algebraic and Analytic Microlocal Analysis" on pointwise Weyl laws for spectral projections kernels in the Kaehler setting. We prove a 2-term pointwise Weyl law for projections onto sums of eigenspaces of spectral width $\hbar=k^{-1}$ of Toeplitz quantizations $\hat{H}_k$ of Hamiltonians on a Kaehler manifold. The first result is a complete asymptotic expansion for smoothed spectral projections in terms of periodic orbit data. When the orbit is `strongly hyperbolic' the leading coefficient defines a uniformly continuous measure on $\R$ and a semi-classical Tauberian theorem implies the 2-term expansion. As in previous works in the series, we use scaling asymptotics of the Boutet-de-Monvel-Sjostrand parametrix and Taylor expansions to reduce the proof to the Bargmann-Fock case.

math.CV↗

arXiv2018

Interface asymptotics of Partial Bergman kernels around a critical level

In a recent series of articles (arXiv:1604.06655, arXiv:1708.09267), the authors have studied the transition behavior of partial Bergman kernels $Π_{k, [E_1, E_2]}(z,w)$ and the associated DOS (density of states) $Π_{k, [E_1, E_2]}(z)$ across the interface $\ccal$ between the allowed and forbidden regions. Partial Bergman kernels are Toeplitz Hamiltonians quantizing Morse functions $H: M \to \R$ on a \kahler manifold. The allowed region is $H^{-1}([E_1, E_2])$ and the interface $\ccal$ is its boundary. In prior articles it was assumed that the endpoints $E_j$ were regular values of $H$. This article completes the series by giving parallel results when an endpoint is a critical value of $H$. In place of the Erf scaling asymptotics in a $k^{-\half} $ tube around $\ccal$ for regular interfaces, one obtains $δ$-asymptotics in $k^{-\frac{1}{4}}$-tubes around singular points of a critical interface. In $k^{-\half}$ tubes, the transition law is given by the osculating metaplectic propagator.

math.CV↗

arXiv2018

Central Limit theorem for toric \kahler manifolds

Associated to the Bergman kernels of a polarized toric \kahler manifold $(M, ω, L, h)$ are sequences of measures $\{μ_k^z\}_{k=1}^{\infty}$ parametrized by the points $z \in M$. For each $z$ in the open orbit, we prove a central limit theorem for $μ_k^z$. The center of mass of $μ_k^z$ is the image of $z$ under the moment map; after re-centering at $0$ and dilating by $\sqrt{k}$, the re-normalized measure tends to a centered Gaussian whose variance is the Hessian of the \kahler potential at $z$. We further give a remainder estimate of Berry-Esseen type. The sequence $\{μ_k^z\}$ is generally not a sequence of convolution powers and the proofs only involve \kahler analysis.

math.PR↗

arXiv2017

Central Limit theorem for spectral Partial Bergman kernels

Partial Bergman kernels $Π_{k, E}$ are kernels of orthogonal projections onto subspaces $\mathcal{k} \subset H^0(M, L^k)$ of holomorphic sections of the $k$th power of an ample line bundle over a Kahler manifold $(M, ω)$. The subspaces of this article are spectral subspaces $\{\hat{H}_k \leq E\}$ of the Toeplitz quantization $\hat{H}_k$ of a smooth Hamiltonian $H: M \to \mathbb{R}$. It is shown that the relative partial density of states $\frac{Π_{k, E}(z)}{Π_k(z)} \to {1}_{\mathcal{A}}$ where $\mathcal{A} = \{H < E\}$. Moreover it is shown that this partial density of states exhibits `Erf'-asymptotics along the interface $\partial \mathcal{A}$, that is, the density profile asymptotically has a Gaussian error function shape interpolating between the values $1,0$ of ${1}_{\mathcal{A}}$. Such `erf'-asymptotics are a universal edge effect. The different types of scaling asymptotics are reminiscent of the law of large numbers and central limit theorem

math.CV↗

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 $e^{- C λ}$

math.AP↗

arXiv2017

Quantum ergodic sequences and equilibrium measures

This is partly a survey article for Constructive Approximation's Special Issue on Approximation and Statistical Physics. It reviews results of B. Shiffman-S.Zelditch, R. Berman. S. Boucksom, D. Witt-Nystrom, T. Bloom, O. Zeitouni and others on Bergman kernels and the asymptotic equilibrium distribution of zeros of random polynomials and more general holomorphic sections of ample line bundles. The article also gives a new definition of `quantum ergodic section of a line bundle' for Hermitian line bundles with general smooth metrics and Bernstein-Markov measurs. It proves the asymptotic equilibrium distribution of zeros of these generalizaed QE sections. It also proves that random sequences and random orthonormal bases of sections are QE.

math.PR↗

arXiv2017

Interface asymptotics of partial Bergman kernels on $S^1$-symmetric Kaehler manifolds

This article is concerned with asymptotics of equivariant Bergman kernels and partial Bergman kernels for polarized projective Kahler manifolds invariant under a Hamiltonian holomorphic $S^1$ action. Asymptotics of partial Bergman kernel are obtained in the allowed region $\mathcal{A}$ resp. forbidden region $\mathcal{F}$, generalizing results of Shiffman-Zelditch, Shiffman-Tate-Zelditch and Pokorny-Singer for toric Kahler manifolds. The main result gives scaling asymptotics of equivariant Bergman kernels and partial Bergman kernels in the transition region around the interface $\partial \mathcal{A}$, generalizing recent work of Ross-Singer on partial Bergman kernels, and refining the Ross-Singer transition asymptotics to apply to equivariant Bergman kernels.

math.CV↗

arXiv2016

Scaling of Harmonic Oscillator Eigenfunctions and Their Nodal Sets Around the Caustic

We study the scaling asymptotics of the eigenspace projection kernels $Π_{\hbar, E}(x,y)$ of the isotropic Harmonic Oscillator $- \hbar ^2 Δ+ |x|^2$ of eigenvalue $E = \hbar(N + \frac{d}{2})$ in the semi-classical limit $\hbar \to 0$. The principal result is an explicit formula for the scaling asymptotics of $Π_{\hbar, E}(x,y)$ for $x,y$ in a $\hbar^{2/3}$ neighborhood of the caustic $\mathcal C_E$ as $\hbar \to 0.$ The scaling asymptotics are applied to the distribution of nodal sets of Gaussian random eigenfunctions around the caustic as $\hbar \to 0$. In previous work we proved that the density of zeros of Gaussian random eigenfunctions of $\hat{H}_{\hbar}$ have different orders in the Planck constant $\hbar$ in the allowed and forbidden regions: In the allowed region the density is of order $\hbar^{-1}$ while it is $\hbar^{-1/2}$ in the forbidden region. Our main result on nodal sets is that the density of zeros is of order $\hbar^{-\frac{2}{3}}$ in an $\hbar^{\frac{2}{3}}$-tube around the caustic. This tube radius is the `critical radius'. For annuli of larger inner and outer radii $\hbar^α$ with $0< α< \frac{2}{3}$ we obtain density results which interpolate between this critical radius result and our prior ones in the allowed and forbidden region. We also show that the Hausdorff $(d-2)$-dimensional measure of the intersection of the nodal set with the caustic is of order $\hbar^{- \frac{2}{3}}$.

math-ph↗

arXiv2016

Heat kernel measures on random surfaces

The heat kernel on the symmetric space of positive definite Hermitian matrices is used to endow the spaces of Bergman metrics of degree k on a Riemann surface M with a family of probability measures depending on a choice of the background metric. Under a certain matrix-metric correspondence, each positive definite Hermitian matrix corresponds to a Kahler metric on M. The one and two point functions of the random metric are calculated in a variety of limits as k and t tend to infinity. In the limit when the time t goes to infinity the fluctuations of the random metric around the background metric are the same as the fluctuations of random zeros of holomorphic sections. This is due to the fact that the random zeros form the boundary of the space of Bergman metrics.

math.PR↗

arXiv2016

Off-diagonal decay of toric Bergman kernels

We study the off-diagonal decay of Bergman kernels $Π_{h^k}(z,w)$ and Berezin kernels $P_{h^k}(z,w)$ for ample invariant line bundles over compact toric projective \kahler manifolds of dimension $m$. When the metric is real analytic, $P_{h^k}(z,w) \simeq k^m \exp - k D(z,w)$ where $D(z,w)$ is the diastasis. When the metric is only $C^{\infty}$ this asymptotic cannot hold for all $(z,w)$ since the diastasis is not even defined for all $(z,w)$ close to the diagonal. We prove that for general $C^{\infty}$ metrics, $P_{h^k}(z,w) \simeq k^m \exp - k D(z,w)$ as long as $w$ lies on the ${\mathbb R}_+^m$-orbit of $z$, and for general $(z,w)$, $\limsup_{k \to \infty} \frac{1}{k} \log P_{h^k}(z,w) \leq - D(z^*,w^*)$ where $D(z, w^*)$ is the diastasis between $z$ and the translate of $w$ by $(S^1)^m$ to the ${\mathbb R}_+^m$ orbit of $z$, complementary to Mike Christ's negative results (arXiv:1308.5644).

math.CV↗