arXiv subjects
Publications and source records attributed to Steve Zelditch.
We prove the convergence of quantized Bergman geodesics to the Mabuchi geodesics for the initial value problem, in the case of real-analytic initial data and in short time. This partially solves a conjecture of Y. Rubinstein and the last author. We also argue against the existence of a solution to the boundary value problem, generically in real-analytic regularity.
The generator of time-translations on the solution space of the wave equation on stationary spacetimes specialises to the square root of the Laplacian on Riemannian manifolds when the spacetime is ultrastatic. Its spectral analysis therefore constitutes a generalization of classical spectral geometry. If the spacetime is spatially compact the spectrum is discrete and admits a wave-trace expansion at time zero. A Weyl law for the eigenvalues and a wave-trace formula was shown in a previous paper and related to the geometry of the space of null-geodesics. In this paper we investigate the relation to heat kernel coefficients and residues of zeta functions in this context and compute the second non-zero term in the wave-trace expansion. This second coefficient is an analogue in the category of stationary spacetimes of the second heat kernel coefficient of the Laplace operator. The general formula is quite involved but reduces to the usual term involving the scalar curvature when specialised to ultra-static spacetimes.
We provide a survey of results on the statistics of random sections of holomorphic line bundles on Kähler manifolds, with an emphasis on the resulting asymptotics when a line bundle is raised to increasing tensor powers. We conclude with a brief discussion of the `Bergman' Kähler metrics induced by these random sections.
Let $\{e_j\}$ be an orthonormal basis of Laplace eigenfunctions of a compact Riemannian manifold $(M,g)$. Let $H \subset M$ be a submanifold and let $\{ψ_k\}$ be an orthonormal basis of Laplace eigenfunctions of $H$ with the induced metric. We obtain joint asymptotics for the Fourier coefficients \[ \langle γ_H e_j, ψ_k \rangle_{L^2(H)} = \int_H e_j \overline ψ_k \, dV_H, \] of restrictions $γ_H e_j$ of $e_j$ to $H$. In particular, we obtain asymptotics for the sums of the norm-squares of the Fourier coefficients over the joint spectrum $\{(μ_k, λ_j)\}_{j,k - 0}^{\infty}$ of the (square roots of the) Laplacian $Δ_M$ on $M$ and the Laplacian $Δ_H$ on $H$ in a family of suitably `thick' regions in $\mathbb R^2$. Thick regions include (1) the truncated cone $μ_k/λ_j \in [a,b] \subset (0,1)$ and $λ_j \leq λ$, and (2) the slowly thickening strip $|μ_k - cλ_j| \leq w(λ)$ and $λ_j \leq λ$, where $w(λ)$ is monotonic and $1 \ll w(λ) \lesssim λ^{1 - 1/n}$. Key tools for obtaining these asymptotics include the composition calculus of Fourier integral operators and a new multidimensional Tauberian theorem.
This article concerns joint asymptotics of Fourier coefficients of restrictions of Laplace eigenfunctions $ϕ_j$ of a compact Riemannian manifold to a submanifold $H \subset M$. We fix a number $c \in (0,1)$ and study the asymptotics of the thin sums, $$ N^{c} _{ε, H }(λ): = \sum_{j, λ_j \leq λ} \sum_{k: |μ_k - c λ_j | < ε} \left| \int_{H} ϕ_j \overline{ψ_k}dV_H \right|^2 $$ where $\{λ_j\}$ are the eigenvalues of $\sqrt{-Δ}_M,$ and $\{(μ_k, ψ_k)\}$ are the eigenvalues, resp. eigenfunctions, of $\sqrt{-Δ}_H$. The inner sums represent the `jumps' of $ N^{c} _{ε, H }(λ)$ and reflect the geometry of geodesic c-bi-angles with one leg on $H$ and a second leg on $M$ with the same endpoints and compatible initial tangent vectors $ξ\in S^c_H M, π_H ξ\in B^* H$, where $π_H ξ$ is the orthogonal projection of $ξ$ to $H$. A c-bi-angle occurs when $\frac{|π_H ξ|}{|ξ|} = c$. Smoothed sums in $μ_k$ are also studied, and give sharp estimates on the jumps. The jumps themselves may jump as $ε$ varies, at certain values of $ε$ related to periodicities in the c-bi-angle geometry. Subspheres of spheres and certain subtori of tori illustrate these jumps. The results refine those of the previous article (arXiv:2011.11571) where the inner sums run over $k: | \frac{μ_k}{λ_j} - c| \leq ε$ and where geodesic bi-angles do not play a role.
We prove that smooth Wigner-Weyl spectral sums at an energy level $E$ exhibit Airy scaling asymptotics across the classical energy surface $Σ_E$. This was proved earlier by the authors for the isotropic harmonic oscillator and the proof is extended in this article to all quantum Hamiltonians $-\hbar^2 Δ+ V$ where $V$ is a confining potential with at most quadratic growth at infinity. The main tools are the Herman-Kluk initial value parametrix for the propagator and the Chester-Friedman-Ursell normal form for complex phases with a one-dimensional cubic degeneracy. This gives a rigorous account of Airy scaling asymptotics of spectral Wigner distributions of M.V. Berry, A. Ozorio de Almeida and other physicists.
We prove that the real parts of equivariant (but non-invariant) eigenfunctions of generic bundle metrics on nontrivial principal $S^1$ bundles over manifolds of any dimension have connected nodal sets and exactly 2 nodal domains. This generalizes earlier results of the authors in the $3$-dimensional case. The failure of the results on for non-free $S^1$ actions is illustrated on even dimensional spheres by one-parameter subgroups of rotations whose fixed point set consists of two antipodal points.
We show that if the eccentricity of an ellipse is sufficiently small then up to isometries it is spectrally unique among all smooth domains. We do not assume any symmetry, convexity, or closeness to the ellipse, on the class of domains. In the course of the proof we also show that for nearly circular domains, the lengths of periodic orbits that are shorter than the perimeter of the domain must belong to the singular support of the wave trace. As a result we also obtain a Laplace spectral rigidity result for the class of axially symmetric nearly circular domains.
This article is about two types of restrictions of eigenfunctions $ϕ_j$ on a compact Riemannian manifold $(M,g)$: First, we restrict to a submanifold $H \subset M$, and expand the restriction $γ_H ϕ_j$ in eigenfunctions $e_k$ of $H$. We then Fourier restrict $γ_H ϕ_j$ to a short interval of eigenvalues of $H$. Laplace eigenvalues of $M$ are denoted $λ_j^2$ and those of $H$ are denoted $μ_k^2$. The Fourier coefficients are negligible unless the $H$- eigenvalues lie in the interval $μ_k \in [-λ_j, λ_j]$. The short windows have the form $|μ_k - c λ_j| < ε$. The goal is to obtain asymptotics and estimates of the Fourier coefficients of $γ_H ϕ_j$ and to see how they vary with $c$. In prior work with E. L. Wyman and Y. Xi, we obtained asymptotics for sums over $(μ_k, λ_j)$ in such windows for $0 < c < 1$. In this article, we obtain `edge' asymptotics when $c=1$ and $H$ is totally geodesic. The order of magnitude and leading coefficient are very different from the case $c<1$. In particular, they depend on the dimension of $H$. We explain how to bridge the bulk results and edge results.
We prove that, under some generic non-degeneracy assumptions, real analytic, centrally symmetric plane domains are determined by their Dirichlet (resp. Neumann) spectra. We prove that the conditions are open-dense for real analytic convex domains. The proof is parallel to the proof that up/down symmetric domains are spectrally determined. One step is to use a Maslov index calculation to show that the second derivative of the defining function of a centrally symmetric domain at the endpoints of a bouncing ball orbit is a spectral invariant. This is also true for up/down symmetric domains, removing an assumption from the proof in that case.
Husimi distributions of Laplace eigenfunctions are special types of `microlocal lifts' of eigenfunctions to phase space. Their weak * limits are the well-known quantum limits or microlocal defect measures of an orthonormal basis $\{ ϕ_j\}$ of eigenfunctions on a Riemannian manifold $(M,g)$ . Husimi distributions are normalized mod squares of analytic continuations of eigenfunctions to the complexification of $M$, which may be identified with an open subset of the cotangent bundle $T^*M$. Husimi distributions are probability measures whose density at $ζ$ is the probability density of a quantum particle at the phase space point $ζ$. We given universal upper bounds on the sup norms of the Husimi distributions. We also give necessary conditions to obtain the upper bounds in terms of the type of the geodesic through $ζ$. The bounds are sharp and are achieved by complexified Gaussian beams. These results open the question of relating sup norms (or other natural norms) of Husimi distributions to properties of the weak * limits.
Associated to the Bergman kernels of a polarized toric Kaehler manifold $(M, ω, L, h)$ are sequences of measures $\{μ_k^z\}_{k=1}^{\infty}$ parametrized by the points $z \in M$. We determine the asymptotics of the entropies $H(μ_k^z)$ of these measures. The sequence $μ_k^z$ in some ways resembles a sequence of convolution powers; we determine precisely when it actually is such a sequence.
A self-focal point of a Riemannian manifold $(M,g)$ is a point $p$ so that every geodesic starting from $p$ returns to $p$ at some positive time. It is called a pole if all geodesics through $p$ are closed, and a non-polar self-focal point if all geodesics loop back but not all are smoothly closed. Umbilic points of two dimensional tri-axial ellipsoids are non-polar self-focal points. Little is known about existence of self-focal points for Riemannian manifolds of dimension $\geq 3$. We prove that ellipsoids of dimension $\geq 3$ with at least 4 distinct axes have no self-focal points. Certain ellipsoids of dimension $\geq 3$ with three distinct axes do have non-polar self-focal points. Ellipsoids with $\leq 2$ distinct axes always have self-focal points. Self-focal points play an important role in the study of $L^{\infty}$ norms of Laplace eigenfunctions. Our results imply that Laplace eigenfunctions on ellipsoids of dimension $\geq 3$ with at least $4$ distinct axes never achieve maximal sup-norm growth.
We give a relativistic generalization of the Gutzwiller-Duistermaat-Guillemin trace formula for the wave group of a compact Riemannian manifold to globally hyperbolic stationary space-times with compact Cauchy hypersurfaces. We introduce several (essentially equivalent) notions of trace of self-adjoint operators on the null-space $\ker \Box$ of the wave operator and define $U(t)$ to be translation by the flow $e^{t Z}$ of the timelike Killing vector field $Z$ on $\Box$. The spectrum of $Z$ on $\ker \Box$ is discrete and the singularities of $\rm{Tr}\;e^{t Z} |_{\ker \Box}$ occur at periods of periodic orbits of $\exp t Z$ on the symplectic manifold of null geodesics. The trace formula gives a Weyl law for the eigenvalues of $Z$ on $\ker \Box$.
We study the spectrum $\{λ_j(m)\}_{j=1}^{\infty}$ of a timelike Killing vector field $Z$ acting as a differential operator $D_Z$ on the Hilbert space of solutions of the massive Klein-Gordon equation $(\Box_g + m^2) u = 0$ on a globally hyperbolic stationary spacetime $(M, g)$ with compact Cauchy hypersurface. The inverse mass $m^{-1}$ is formally like the Planck constant in a Schrödinger equation, and we give Weyl asymptotics as $m \to \infty$ for the number $$N_{ν, C}(m)= \# \{j \mid \frac{λ_j(m)}{m} \in [ν- \frac{C}{m}, ν+ \frac{C}{m} ]\}$$ for a given $C > 0$. The semi-classical mass asymptotics are governed by the dynamics of the Killing flow $e^{tZ} $ on the hypersurface in the space of mass $1$ geodesics $γ$ where $\langle \dotγ, Z \rangle= ν$.
This is largely a survey of results obtained jointly with Boris Hanin and Peng Zhou on interfaces in spectral asymptotics, both for Schrödinger operators on $L^2({\mathbb R}^d)$ and for Toeplitz Hamiltonians acting on holomorphic sections of ample line bundles $L \to M$ over Kähler manifolds $(M, ω)$. By an interface is meant a hypersurface, either in physical space ${\mathbb R}^d$ or in phase space, separating an allowed region where spectral asymptotics are standard and a forbidden region where they are non-standard. The main question is to give the detailed transition between the two types of asymptotics across the hypersurface (i.e. interface). In the real Schrödinger setting, the asymptotics are of Airy type; in the Kähler setting they are of Erf (Gaussian error function) type. In addition, we introduce the Bargmann-Fock space of a positive Hermitian line bundle and study interface asymptotics in that setting.
This paper is part of a series concerning the isospectral problem for an ellipse. In this paper, we study Cauchy data of eigenfunctions of the ellipse with Dirichlet or Neumann boundary conditions. Using many classical results on ellipse eigenfunctions, we determine the microlocal defect measures of the Cauchy data of the eigenfunctions. The ellipse has integrable billiards, i.e. the boundary phase space is foliated by invariant curves of the billiard map. We prove that, for any invariant curve $C$, there exists a sequence of eigenfunctions whose Cauchy data concentrates on $C$. We use this result to give a new proof that ellipses are infinitesimally spectrally rigid among $C^{\infty}$ domains with the symmetries of the ellipse.
We prove a uniform lower bound on Cauchy data on an arbitrary curve on a negatively curved surface using the Dyatlov-Jin(-Nonnenmacher) observability estimate on the global surface. In the process, we prove some further results about defect measures of restrictions of eigenfunctions to a hypersurface.