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Steve Zelditch

Publications and source records attributed to Steve Zelditch.

At least 73 records · Page 4Linked to original sources
arXiv2013

Matrix elements of Fourier Integral Operators

This article is concerned with the semi-classical limits of matrix elements $ $ of eigenfunctions of the Laplacian $Δ_g$ of a compact Riemannian manifold $(M, g)$ with respect to a Fourier integral operator $F$ on $L^2(M)$. Many results exist for the case where $F$ is a pseudo-differential operator, but matrix elements of Fourier integral operators involve new considerations. The limits reflect the extent to which the canonical relation of $F$ is invariant under the geodesic flow of $(M, g)$. When the canonical relation is almost nowhere invariant, a density one subsequence of the matrix elements tends to zero (related results arose first in the study of quantum ergodic restriction theorems). The limit states are invariant measures on the canonical relation of $F$ and their invariance properties are explained. The invariance properties in the case of Hecke operators answers an old question raised by the author.

math.SP↗
arXiv2013

Quantum ergodic restriction for Cauchy data: Interior QUE and restricted QUE

We prove a quantum ergodic restriction theorem for the Cauchy data of a sequence of quantum ergodic eigenfunctions on a hypersurface $H$ of a Riemannian manifold $(M, g)$. The technique of proof is to use a Rellich type identity to relate quantum ergodicity of Cauchy data on $H$ to quantum ergodicity of eigenfunctions on the global manifold $M$. This has the interesting consequence that if the eigenfunctions are quantum unique ergodic on the global manifold $M$, then the Cauchy data is automatically quantum unique ergodic on $H$ with respect to operators whose symbols vanish to order one on the glancing set of unit tangential directions to $H$.

math.AP↗
arXiv2013

Median and mean of the Supremum of $L^2$ normalized random holmorphic fields

We prove that the expected value and median of the supremum of $L^2$ normalized random holomorphic fields of degree $n$ on $m$-dimensional Kähler manifolds are asymptotically of order $\sqrt{m\log n}$. This improves the prior result of Shiffman-Zelditch (arXiv:math/0303335) that the upper bound of the media is of order $\sqrt{\log n}$ The estimates are based on the entropy methods of Dudley and Sudakov combined with a precise analysis of the relevant pseudo-metric and its covering numbers, which can be precisely evaluated using off-diagonal asymptotics of Bergman kernels. Recent work of the authors on the value distribution of these fields are also used to get precise constants.

math.PR↗
arXiv2013

Random Kähler Metrics

The purpose of this article is to propose a new method to define and calculate path integrals over metrics on a Kähler manifold. The main idea is to use finite dimensional spaces of Bergman metrics, as an approximation to the full space of Kähler metrics. We use the theory of large deviations to decide when a sequence of probability measures on the spaces of Bergman metrics tends to a limit measure on the space of all Kähler metrics. Several examples are considered.

hep-th↗
arXiv2012

Lower bounds on the Hausdorff measure of nodal sets II

We give a very short argument showing how the main identity from our earlier paper (Sogge and Zelditch, 2011) immediately leads to the best lower bound currently known (Colding and Minicozzi II, 2011) for the Hausdorff measure of nodal sets in dimensions $n\ge 3$.

math.AP↗
arXiv2012

Critical values of random analytic functions on complex manifolds

We study the asymptotic distribution of critical values of random holomorphic `polynomials' s_n on a Kaehler manifold M as the degree n tends to infinity. By `polynomial' of degree n we mean a holomorphic section of the nth power of a positive Hermitian holomorphic line bundle $(L, h). In the special case M = CP^m and L = O(1), and h is the Fubini-Study metric, the random polynomials are the SU(m + 1) polynomials. By a critical value we mean the norm ||s_n||_h of s_n at a non-zero critical point of the norm. The metric and Kahler form endow the polynomials with a Hilbert space structure and we consider the associated Gaussian random polynomials and the spherical ensemble where ||s_n|| = 1 is chosen from Haar measure. Our main result is that the expected limit distribution of critical values as n tends to infinity in the spherical ensemble is universal (i.e. is independent of the choice of h), and we give an explicit formula for it. The limit distribution is the same as for suitably normalized Gaussian measures.

math.PR↗
arXiv2012

Random orthonormal bases of spaces of high dimension

We consider a sequence H_N of Hilbert spaces of dimensions d_N tending to infinity. The motivating examples are eigenspaces or quasi-mode spaces of a Laplace or Schrodinger operator. We define a random ONB of H_N by fixing one ONB and changing it by a random element of U(d_N). A random ONB of the direct sum of the H_N is an independent sequence {U_N} of random ONB's of the H_N. We prove that if d_N tends to infinity and if the normalized traces of observables in H_N tend to a unique limit state, then a random ONB also tends to that limit state. This generalizes an earlier result of the author for eigenspaces of the standard 2-sphere, and shows that the result does not depend on how fast the dimensions grow. In particular it is valid for eigenspaces of a flat rational torus in dimensions > 4. The main idea is to convert quantum ergodicity into a problem on the moments of inertia of permutahedra and to calculate the moments using Schur polynomials.

math.SP↗
arXiv2012

Ergodicity and intersections of nodal sets and geodesics on real analytic surfaces

We consider the the intersections of the complex nodal set of the analytic continuation of an eigenfunction of the Laplacian on a real analytic surface with the complexification of a geodesic. We prove that if the geodesic flow is ergodic and if the geodesic is periodic and satisfies a generic asymmetry condition, then the intersection points condense along the real geodesic and become uniformly distributed with respect to its arc-length. We prove an analogous result for non-periodic geodesics except that the `origin' is allowed to move with $λ_j$. The proof uses the quantum ergodic restriction theorem due to J. Toth and the author (see also Dyatlov-Zworski).

math.SP↗
arXiv2012

The Cauchy problem for the homogeneous Monge-Ampere equation, III. Lifespan

We prove several results on the lifespan, regularity, and uniqueness of solutions of the Cauchy problem for the homogeneous complex and real Monge-Ampere equations (HCMA/HRMA) under various a priori regularity conditions. We use methods of characteristics in both the real and complex settings to bound the lifespan of solutions with prescribed regularity. In the complex domain, we characterize the C^3 lifespan of the HCMA in terms of analytic continuation of Hamiltonian mechanics and intersection of complex time characteristics. We use a conservation law type argument to prove uniqueness of solutions of the Cauchy problem for the HCMA. We then prove that the Cauchy problem is ill-posed in C^3, in the sense that there exists a dense set of C^3 Cauchy data for which there exists no C^3 solution even for a short time. In the real domain we show that the HRMA is equivalent to a Hamilton--Jacobi equation, and use the equivalence to prove that any differentiable weak solution is smooth, so that the differentiable lifespan equals the convex lifespan determined in our previous articles. We further show that the only obstruction to C^1 solvability is the invertibility of the associated Moser maps. Thus, a smooth solution of the Cauchy problem for HRMA exists for a positive but generally finite time and cannot be continued even as a weak C^1 solution afterwards. Finally, we introduce the notion of a "leafwise subsolution" for the HCMA that generalizes that of a solution, and many of our aforementioned results are proved for this more general object.

math.DG↗
arXiv2012

Simple matrix models for random Bergman metrics

Recently, the authors have proposed a new approach to the theory of random metrics, making an explicit link between probability measures on the space of metrics on a Kahler manifold and random matrix models. We consider simple examples of such models and compute the one and two-point functions of the metric. These geometric correlation functions correspond to new interesting types of matrix model correlators. We study a large class of examples and provide in particular a detailed study of the Wishart model.

hep-th↗
arXiv2011

Random Riesz energies on compact Kähler manifolds

This article determines the asymptotics of the expected Riesz s-energy of the zero set of a Gaussian random systems of polynomials of degree N as the degree N tends to infinity in all dimensions and codimensions. The asymptotics are proved more generally for sections of any positive line bundle over any compact Kaehler manifold. In comparison with the results on energies of zero sets in one complex dimension due to Qi Zhong (arXiv:0705.2000) (see also [arXiv:0705.2000]), the zero sets have higher energies than randomly chosen points in dimensions > 2 due to clumping of zeros.

math.PR↗
arXiv2011

Gravitational Actions in Two Dimensions and the Mabuchi Functional

The Mabuchi energy is an interesting geometric functional on the space of Kähler metrics that plays a crucial rôle in the study of the geometry of Kähler manifolds. We show that this functional, as well as other related geometric actions, contribute to the effective gravitational action when a massive scalar field is coupled to gravity in two dimensions in a small mass expansion. This yields new theories of two-dimensional quantum gravity generalizing the standard Liouville models.

hep-th↗
arXiv2011

On eigenfunction restriction estimates and $L^4$-bounds for compact surfaces with nonpositive curvature

Let $(M,g)$ be a two-dimensional compact boundaryless Riemannian manifold with nonpostive curvature, then we shall give improved estimates for the $L^2$-norms of the restrictions of eigenfunctions to unit-length geodesics, compared to the general results of Burq, Gérard and Tzvetkov \cite{burq}. By earlier results of Bourgain \cite{bourgainef} and the first author \cite{Sokakeya}, they are equivalent to improvements of the general $L^p$-estimates in \cite{soggeest} for $n=2$ and $2<p<6$. The proof uses the fact that the exponential map from any point in $x_0\in M$ is a universal covering map from $\Rt \simeq T_{x_0}M$ to $M$ (the Cartan-Hadamard- von Mangolt theorem), which allows us to lift the necessary calculations up to the universal cover $(\Rt, \tilde g)$ where $\tilde g$ is the pullback of $g$ via the exponential map. We then prove the main estimates by using the Hadamard parametrix for the wave equation on $(\Rt, \tilde g)$ and the fact that the classical comparison theorem of Günther \cite{Gu} for the volume element in spaces of nonpositive curvature gives us desirable bounds for the principal coefficient of the Hadamard parametrix, allowing us to prove our main result.

math.AP↗
arXiv2011

Random Geometry, Quantum Gravity and the Kähler Potential

We propose a new method to define theories of random geometries, using an explicit and simple map between metrics and large hermitian matrices. We outline some of the many possible applications of the formalism. For example, a background-independent measure on the space of metrics can be easily constructed from first principles. Our framework suggests the relevance of a new gravitational effective action and we show that it occurs when coupling the massive scalar field to two-dimensional gravity. This yields new types of quantum gravity models generalizing the standard Liouville case.

hep-th↗
arXiv2011

Pluri-potential theory on Grauert tubes of real analytic Riemannian manifolds, I

We develop analogues for Grauert tubes of real analytic Riemannian manifolds (M,g) of some basic notions of pluri-potential theory, such as the Siciak extremal function. The basic idea is to use analytic continuations of eigenfunctions in place of polynomials or sections of powers of positive line bundles for pluripotential theory. The analytically continued Poisson-wave kernel plays the role of Bergman kernel. The main results are Weyl laws in the complex domain, distribution of complex zeros of eigenfunctions on locally symmetric spaces, and estimates of triple products of eigenfunctions.

math.SP↗
arXiv2011

Quantum ergodic restriction theorems, I: interior hypersurfaces in domains with ergodic billiards

Quantum ergodic restriction (QER) is the problem of finding conditions on a hypersurface $H$ so that restrictions $ϕ_j |_H$ to $H$ of $Δ$-eigenfunctions of Riemannian manifolds $(M, g)$ with ergodic geodesic flow are quantum ergodic on $H$. We prove two kinds of results: First (i) for any smooth hypersurface $H$, the Cauchy data $(ϕ_j|H, \partial ϕ_j|H)$ is quantum ergodic if the Dirichlet and Neumann data are weighted appropriately. Secondly (ii) we give conditions on $H$ so that the Dirichlet (or Neumann) data is individually quantum ergodic. The condition involves the almost nowhere equality of left and right Poincaré maps for $H$. The proof involves two further novel results: (iii) a local Weyl law for boundary traces of eigenfunctions, and (iv) an 'almost-orthogonality' result for Fourier integral operators whose canonical relations almost nowhere commute with the geodesic flow.

math.AP↗
arXiv2010

Lower bounds on the Hausdorff measure of nodal sets

Let $\ncal_{ϕ_λ}$ be the nodal hypersurface of a $Δ$-eigenfunction $ϕ_λ$ of eigenvalue $λ^2$ on a smooth Riemannian manifold. We prove the following lower bound for its surface measure: $\hcal^{n-1}(\ncal_{ϕ_λ}) \geq C λ^{\frac74-\frac{3n}4} $. The best prior lower bound appears to be $e^{- C λ}$.

math.AP↗