Quantum Ergodicity for Eisenstein functions
A new proof is given of Quantum Ergodicity for Eisenstein Series for cusped hyperbolic surfaces. This result is also extended to higher dimensional examples, with variable curvature.
math.AP↗
arXiv subjects
Steve Zelditch
Publications and source records attributed to Steve Zelditch.
Completeness of boundary traces of eigenfunctions
In this paper, we study the boundary traces of eigenfunctions on the boundary of a smooth and bounded domain. An identity derived by Bäcker, Fürstburger, Schubert, and Steiner, expressing (in some sense) the asymptotic completeness of the set of boundary traces in a frequency window of size O(1), is proved both for Dirichlet and Neumann boundary conditions. We then prove a semiclassical generalization of this identity.
math.AP↗
Logarithmic lower bound on the number of nodal domains
We prove that the number of nodal domains of a density one subsequence of eigenfunctions grows at least logarithmically with the eigenvalue on negatively curved `real Riemann surfaces'. The geometric model is the same as in prior joint work with Junehyuk Jung (arXiv:1310.2919, to appear in J. Diff. Geom), where the number of nodal domains was shown to tend to infinity, but without a specified rate. The proof of the logarithmic rate uses the new logarithmic scale quantum ergodicity results of Hezari-Riviere (arXiv:1411.4078) and X. Han (arXiv:1410.3911).
math.SP↗
Survey of the inverse spectral problem for the Notices of the ICCM
This is an updated survey on the inverse spectral problem written for the Notices of the ICCM. It rapidly reviews some of the material in the previous survey of the same title (arXiv:math/0402356) and then discusses some relatively new results (rigidity results for the ellipse, phase shifts in scattering theory). The last section poses a number of open problems involving "inverse results for oscillatory integrals".
math.SP↗
Critical values of fixed Morse index of random analytic functions on Riemann surfaces
This note is an addendum to 'Critical values of random analytic functions on complex manifolds, Indiana Univ. Math. J. 63 No. 3 (2014), 651-686.' by R.Feng and S. Zelditch (arXiv:1212.4762). In this note, we give the formula of the limiting distribution of the critical values at critical points of random analytic functions of fixed Morse index and we compute explicitly such limitings for local maxima and saddle values in the case of Riemann surfaces; we also prove that the second order term in the full expansion of the distribution of the critical values is topologically invariant on Riemann surfaces.
math.PR↗
Number of nodal domains of eigenfunctions on non-positively curved surfaces with concave boundary
It is an open problem in general to prove that there exists a sequence of $Δ_g$-eigenfunctions $ϕ_{j_k}$ on a Riemannian manifold $(M, g)$ for which the number $N(ϕ_{j_k}) $ of nodal domains tends to infinity with the eigenvalue. Our main result is that $N(ϕ_{j_k}) \to \infty$ along a subsequence of eigenvalues of density $1$ if the $(M, g)$ is a non-positively curved surface with concave boundary, i.e. a generalized Sinai or Lorentz billiard. Unlike the recent closely related work of Ghosh-Reznikov-Sarnak and of the authors on the nodal domain counting problem, the surfaces need not have any symmetries.
math.SP↗
Focal points and sup-norms of eigenfunctions on analytic Riemannian manifolds II: the two-dimensional case
In the recent work arXiv:1311.3999, the authors proved that real analytic manifolds $(M, g)$ with maximal eigenfunction growth must have a self-focal point p whose first return map has an invariant L1 measure on $S^*_p M$. In this addendum we add a purely dynamical argument on circle maps to improve the conclusion to: all geodesics from p are smoothly closed.
math.SP↗
Scaling asymptotics of heat kernels of line bundles
We consider a general Hermitian holomorphic line bundle $L$ on a compact complex manifold $M$ and let ${\Box}^q_p$ be the Kodaira Laplacian on $(0,q)$ forms with values in $L^p$. The main result is a complete asymptotic expansion for the semi-classically scaled heat kernel $\exp(-u{\Box}^q_p/p)(x,x)$ along the diagonal. It is a generalization of the Bergman/Szegö kernel asymptotics in the case of a positive line bundle, but no positivity is assumed. We give two proofs, one based on the Hadamard parametrix for the heat kernel on a principal bundle and the second based on the analytic localization of the Dirac-Dolbeault operator.
math.CV↗
Szego kernels and Poincare series
Let $M = \tilde{M}/Γ$ be a Kahler manifold, where $\tilde{M}$ is the universal Kahler cover, and where $Γ$ is the deck transformation group. Let $(L, h) \to M$ be a positive Hermitian holomorophic line bundle. Lift the Hermitian line bundle to $\tilde{M}$ and consider the relation between the orthogonal projection onto holomorphic sections on the quotient and onto $L^2$ holomorphic sections on $\tilde{M}$. We prove that the quotient Szego kernel is given by the periodization of the $L^2$ Szego kernel of the universal cover, i.e. its sum over the deck group $Γ$. Although this is a standard result for symmetric spaces (it is used in the Selberg trace formula) and is also standard for heat and wave kernels, it seems that a proof of the sum over Gamma formula was lacking for Szego kernels of positive line bundles over general Kahler manifolds. We apply the result to give a simple proof of Napier's theorem on the holomorphic convexity of $\tilde{M}$ with respect to $\tilde{L}^N$ and to surjectivity of Poincaré series. We also simplify the discussion of Poincare series in Kollar's book.
math.CV↗
Gaussian beams on Zoll manifolds and maximally degenerate Laplacians
Gaussian beams exist along all closed geodesics of a Zoll surface, despite the fact that the algorithm for constructing them assumes that the closed geodesics are non-degenerate. Similarly, there exists a global Birkhoff normal for a Zoll Laplacian despite the degeneracy. We explain why both algorithms work in the Zoll case and give an exact formula for the sub-principal normal form invariant. In the case of "maximally degenerate" Zoll Laplacians, this invariant vanishes and we obtain new geometric constraints on such Zoll metrics.
math.SP↗
Stability and integration over Bergman metrics
We study partition functions of random Bergman metrics, with the actions defined by a class of geometric functionals known as `stability functions'. We introduce a new stability invariant - the critical value of the coupling constant - defined as the minimal coupling constant for which the partition function converges. It measures the minimal degree of stability of geodesic rays in the space the Bergman metrics, with respect to the action. We calculate this critical value when the action is the $ν$-balancing energy, and show that $γ_k^{\rm crit} = k - h$ on a Riemann surface of genus $h$.
hep-th↗
Measure of nodal sets of analytic Steklov eigenfunctions
Let $(Ω, g)$ be a real analytic Riemannian manifold with real analytic boundary $\partial Ω$. Let $ψ_λ$ be an eigenfunction of the Dirichlet-to-Neumann operator $Λ$ of $(Ω, g, \partial Ω)$ of eigenvalue $λ$. Let $\mathcal N_{λ_j}$ be its nodal set. Then $\mathcal H^{n-2} (\mathcal N_λ) \leq C_{g, Ω} λ.$ This proves a conjecture of F. H. Lin and K. Bellova.
math.SP↗
A note on $L^p$-norms of quasi-modes
In this note we show how improved $L^p$-estimates for certain types of quasi-modes are naturally equaivalent to improved operator norms of spectral projection operators associated to shrinking spectral intervals of the appropriate scale. Using this, one can see that recent estimates that were stated for eigenfunctions also hold for the appropriate types of quasi-modes.
math.AP↗
Focal points and sup-norms of eigenfunctions
If $(M,g)$ is a compact real analytic Riemannian manifold, we give a necessary and sufficient condition for there to be a sequence of quasimodes of order $o(λ)$ saturating sup-norm estimates. In particular, it gives optimal conditions for existence of eigenfunctions satisfying maximal sup norm bounds. The condition is that there exists a self-focal point $x_0\in M$ for the geodesic flow at which the associated Perron-Frobenius operator $U_{x_0}: L^2(S_{x_0}^*M) \to L^2(S_{x_0}^*M)$ has a nontrivial invariant $L^2$ function. The proof is based on an explict Duistermaat-Guillemin-Safarov pre-trace formula and von Neumann's ergodic theorem.
math.AP↗
Nodal Sets of Random Eigenfunctions for the Isotropic Harmonic Oscillator
We consider Gaussian random eigenfunctions (Hermite functions) of fixed energy level of the isotropic semi-classical Harmonic Oscillator on ${\bf R}^n$. We calculate the expected density of zeros of a random eigenfunction in the semi-classical limit $h \to 0.$ In the allowed region the density is of order $h^{-1},$ while in the forbidden region the density is of order $h^{-\frac{1}{2}}$. The computer graphics due to E.J. Heller illustrate this difference in "frequency" between the allowed and forbidden nodal sets.
math.PR↗
Equidistribution of phase shifts in semiclassical potential scattering
Consider a semiclassical Hamiltonian $H := h^{2} Δ+ V - E$ where $Δ$ is the positive Laplacian on $\mathbb{R}^{d}$, $V \in C^{\infty}_{0}(\mathbb{R}^{d})$ and $E > 0$ is an energy level. We prove that under an appropriate dynamical hypothesis on the Hamilton flow corresponding to $H$, the eigenvalues of the scattering matrix $S_{h}(V)$ define a measure on $\mathbb{S}^{1}$ that converges to Lebesgue measure away from $1 \in \mathbb{S}^{1}$ as $h \to 0$.
math.SP↗
Park City lectures on Eigenfunctions
This is an expository article based on my lectures on eigenfunctions of the Laplacian for the 2013 IAS/Park City Mathematics Institute (PCMI) summer school in geometric analysis. Many of the results are based on joint work with H. Christianson, J. Jung, C. Sogge, and J. Toth.
math.AP↗
Number of nodal domains and singular points of eigenfunctions of negatively curved surfaces with an isometric involution
We prove two types of nodal results for density one subsequences of an orthonormal basis $\{ϕ_j\}$ of eigenfunctions of the Laplacian on a negatively curved compact surface. The first type of result involves the intersections $Z_{ϕ_j} \cap H$ of the nodal set $Z_{ϕ_j}$ of $ϕ_j$ with a smooth curve $H$. Using recent results on quantum ergodic restriction theorems and prior results on periods of eigenfunctions over curves, we prove that under an asymmetry assumption on $H$, the number of intersection points tends to infinity for a density one subsequence of the $ϕ_j$. . We also prove that the number of zeros of the normal derivative $\partial_ν ϕ_j$ on $H$ tends to infinity. From these results we obtain a lower bound on the number of nodal domains of even and odd eigenfunctions on surfaces with an isometric involution. Using (and generalizing) a geometric argument of Ghosh-Reznikov-Sarnak, we show that the number of nodal domains of even or odd eigenfunctions tends to infinity for a density one subsequence of eigenfunctions.
math.SP↗