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Jun 28, 2018
06/18
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Steve Zelditch
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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...
Topics: Spectral Theory, Mathematics
Source: http://arxiv.org/abs/1510.05315
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Sep 21, 2013
09/13
by
Christopher D. Sogge; Steve Zelditch
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Let $(M,g)$ be a two-dimensional compact boundaryless Riemannian manifold with Laplacian, $\Delta_g$. If $e_\lambda$ are the associated eigenfunctions of $\sqrt{-\Delta_g}$ so that $-\Delta_g e_\lambda = \lambda^2 e_\lambda$, then it has been known for some time \cite{soggeest} that $\|e_\lambda\|_{L^4(M)}\lesssim \lambda^{1/8}$, assuming that $e_\lambda$ is normalized to have $L^2$-norm one. This result is sharp in the sense that it cannot be improved on the standard sphere because of highest...
Source: http://arxiv.org/abs/1011.0215v1
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Jun 29, 2018
06/18
by
Boris Hanin; Steve Zelditch; Peng Zhou
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We study the scaling asymptotics of the eigenspace projection kernels $\Pi_{\hbar, E}(x,y)$ of the isotropic Harmonic Oscillator $- \hbar ^2 \Delta + |x|^2$ 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 $\Pi_{\hbar, E}(x,y)$ 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...
Topics: Probability, Spectral Theory, Mathematical Physics, Mathematics
Source: http://arxiv.org/abs/1602.06848
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3.0
Jun 30, 2018
06/18
by
Chris Sogge; Steve Zelditch
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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.
Topics: Mathematics, Spectral Theory, Analysis of PDEs
Source: http://arxiv.org/abs/1409.2063
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Sep 19, 2013
09/13
by
John A. Toth; Steve Zelditch
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The eigenfunctions e^{i \lambda x} of the Laplacian on a flat torus have uniformly bounded L^p norms. In this article, we prove that for every other quantum integrable Laplacian, the L^p norms of the joint eigenfunctions must blow up at a rate \gg \lambda^{p-2/4p - \epsilon} for every \epsilon >0 as \lambda \to \infty.
Source: http://arxiv.org/abs/math/0208045v1
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Sep 19, 2013
09/13
by
Jian Song; Steve Zelditch
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The space of positively curved hermitian metrics on a positive holomorphic line bundle over a compact complex manifold is an infinite-dimensional symmetric space. It is shown by Phong and Sturm that geodesics in this space can be uniformly approximated by geodesics in the finite dimensional spaces of Bergman metrics. We prove a stronger C^2-approximation in the special case of toric (i.e. S^1-invariant) metrics on CP^1.
Source: http://arxiv.org/abs/math/0703517v1
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Sep 19, 2013
09/13
by
Christopher D. Sogge; Steve Zelditch
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Let $\ncal_{\phi_{\lambda}}$ be the nodal hypersurface of a $\Delta$-eigenfunction $\phi_{\lambda}$ of eigenvalue $\lambda^2$ on a smooth Riemannian manifold. We prove the following lower bound for its surface measure: $\hcal^{n-1}(\ncal_{\phi_{\lambda}}) \geq C \lambda^{\frac74-\frac{3n}4} $. The best prior lower bound appears to be $e^{- C \lambda}$.
Source: http://arxiv.org/abs/1009.3573v3
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Sep 20, 2013
09/13
by
Yanir A. Rubinstein; Steve Zelditch
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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...
Source: http://arxiv.org/abs/1205.4793v1
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Sep 19, 2013
09/13
by
Renjie Feng; Steve Zelditch
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We extend results of Zeitouni-Zelditch on large deviations principles for zeros of Gaussian random polynomials $s$ in one complex variable to certain non-Gaussian ensembles that we call $P(\phi)_2$ random polynomials. The probability measures are of the form $e^{- S(f)} df$ where the actions $S(f)$ are finite dimensional analgoues of those of $P(\phi)_2$ quantum field theory. The speed and rate function are the same as in the associated Gaussian case. As a corollary, we prove that the expected...
Source: http://arxiv.org/abs/1009.5142v1
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Jun 27, 2018
06/18
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Semyon Klevtsov; Steve Zelditch
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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...
Topics: High Energy Physics - Theory, Complex Variables, Mathematics, Probability
Source: http://arxiv.org/abs/1505.05546
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5.0
Jun 30, 2018
06/18
by
Junehyuk Jung; Steve Zelditch
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It is an open problem in general to prove that there exists a sequence of $\Delta_g$-eigenfunctions $\phi_{j_k}$ on a Riemannian manifold $(M, g)$ for which the number $N(\phi_{j_k}) $ of nodal domains tends to infinity with the eigenvalue. Our main result is that $N(\phi_{j_k}) \to \infty$ 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...
Topics: Mathematics, Spectral Theory, Analysis of PDEs
Source: http://arxiv.org/abs/1401.4520
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Sep 20, 2013
09/13
by
Nalini Anantharaman; Steve Zelditch
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We relate two types of phase space distributions associated to eigenfunctions $\phi_{ir_j}$ of the Laplacian on a compact hyperbolic surface $X_{\Gamma}$: (1) Wigner distributions $\int_{S^*\X} a dW_{ir_j}= < Op(a)\phi_{ir_j}, \phi_{ir_j}>_{L^2(\X)}$, which arise in quantum chaos. They are invariant under the wave group. (2) Patterson-Sullivan distributions $PS_{ir_j}$, which are the residues of the dynamical zeta-functions $\lcal(s; a): = \sum_\gamma...
Source: http://arxiv.org/abs/math/0601776v2
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Sep 18, 2013
09/13
by
Steve Zelditch
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It does not seem to have been observed previously that the classical Bernstein polynomials $B_N(f)(x)$ are closely related to the Bergman-Szego kernels $\Pi_N$ for the Fubini-Study metric on $\CP^1$: $B_N(f)(x)$ is the Berezin symbol of the Toeplitz operator $\Pi_N f(N^{-1} D_{\theta})$. The relation suggests a generalization of Bernstein polynomials to any toric Kahler variety and Delzant polytope $P$. When $f$ is smooth, $B_N(f)(x)$ admits a complete asymptotic expansion. Integrating it over...
Source: http://arxiv.org/abs/0705.2879v3
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Sep 22, 2013
09/13
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Christopher D. Sogge; John A. Toth; Steve Zelditch
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On any compact Riemannian manifold $(M, g)$ of dimension $n$, the $L^2$-normalized eigenfunctions ${\phi_{\lambda}}$ satisfy $||\phi_{\lambda}||_{\infty} \leq C \lambda^{\frac{n-1}{2}}$ where $-\Delta \phi_{\lambda} = \lambda^2 \phi_{\lambda}.$ The bound is sharp in the class of all $(M, g)$ since it is obtained by zonal spherical harmonics on the standard $n$-sphere $S^n$. But of course, it is not sharp for many Riemannian manifolds, e.g. flat tori $\R^n/\Gamma$. We say that $S^n$, but not...
Source: http://arxiv.org/abs/0908.0688v1
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Sep 18, 2013
09/13
by
Steve Zelditch
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We consider Riemannian random waves, i.e. Gaussian random linear combination of eigenfunctions of the Laplacian on a compact Riemannian manifold with frequencies from a short interval (`asymptotically fixed frequency'). We first show that the expected limit distribution of the real zero set of a is uniform with respect to the volume form of a compact Riemannian manifold $(M, g)$. We then show that the complex zero set of the analytic continuations of such Riemannian random waves to a Grauert...
Source: http://arxiv.org/abs/0803.4334v1
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6.0
Jun 28, 2018
06/18
by
Yannick Bonthonneau; Steve Zelditch
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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.
Topics: Analysis of PDEs, Spectral Theory, Mathematics
Source: http://arxiv.org/abs/1512.06802
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Jul 20, 2013
07/13
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Frank Ferrari; Semyon Klevtsov; Steve Zelditch
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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...
Source: http://arxiv.org/abs/1107.4022v2
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Sep 23, 2013
09/13
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Steve Zelditch
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This is a survey on eigenfunctions of the Laplacian on Riemannian manifolds (mainly compact and without boundary). We discuss both local results obtained by analyzing eigenfunctions on small balls, and global results obtained by wave equation methods. Among the main topics are nodal sets, quantum limits, and $L^p$ norms of global eigenfunctions.
Source: http://arxiv.org/abs/0903.3420v1
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Jul 20, 2013
07/13
by
Steve Zelditch
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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...
Source: http://arxiv.org/abs/1107.0463v1
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Sep 18, 2013
09/13
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Steve Zelditch
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We give a simple proof of Tian's theorem that the Kodaira embeddings associated to a positive line bundle over a compact complex manifold are asymptotically isometric. The proof is based on the diagonal asymptotics of the Szego kernel (i.e. the orthogonal projection onto holomorphic sections). In deriving these asymptotics we use the Boutet de Monvel-Sjostrand parametrix for the Szego kernel.
Source: http://arxiv.org/abs/math-ph/0002009v1
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Sep 20, 2013
09/13
by
Bernard Shiffman; Steve Zelditch
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We show that the variance of the number of simultaneous zeros of $m$ i.i.d. Gaussian random polynomials of degree $N$ in an open set $U \subset C^m$ with smooth boundary is asymptotic to $N^{m-1/2} \nu_{mm} Vol(\partial U)$, where $\nu_{mm}$ is a universal constant depending only on the dimension $m$. We also give formulas for the variance of the volume of the set of simultaneous zeros in $U$ of $k
Source: http://arxiv.org/abs/math/0608743v3
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Sep 24, 2013
09/13
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Frank Ferrari; Semyon Klevtsov; Steve Zelditch
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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...
Source: http://arxiv.org/abs/1112.4382v2
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Jul 20, 2013
07/13
by
Michael R. Douglas; Bernard Shiffman; Steve Zelditch
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Motivated by the vacuum selection problem of string/M theory, we study a new geometric invariant of a positive Hermitian line bundle $(L, h)\to M$ over a compact K\"ahler manifold: the expected distribution of critical points of a Gaussian random holomorphic section $s \in H^0(M, L)$ with respect to the Chern connection $\nabla_h$. It is a measure on $M$ whose total mass is the average number $\mathcal{N}^{crit}_h$ of critical points of a random holomorphic section. We are interested in...
Source: http://arxiv.org/abs/math/0406089v3
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6.0
Jun 30, 2018
06/18
by
Christopher D. Sogge; Steve Zelditch
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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.
Topics: Mathematics, Analysis of PDEs, Classical Analysis and ODEs
Source: http://arxiv.org/abs/1401.0345
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Sep 19, 2013
09/13
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Yanir A. Rubinstein; Steve Zelditch
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We continue our study of the Cauchy problem for the homogeneous (real and complex) Monge-Ampere equation (HRMA/HCMA). In the prequel a quantum mechanical approach for solving the HCMA was developed, and was shown to coincide with the well-known Legendre transform approach in the case of the HRMA. In this article---that uses tools of convex analysis and can be read independently---we prove that the candidate solution produced by these methods ceases to solve the HRMA, even in a weak sense, as...
Source: http://arxiv.org/abs/1010.2442v2
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Sep 21, 2013
09/13
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Bernard Shiffman; Steve Zelditch
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We introduce several notions of `random fewnomials', i.e. random polynomials with a fixed number f of monomials of degree N. The f exponents are chosen at random and then the coefficients are chosen to be Gaussian random, mainly from the SU(m + 1) ensemble. The results give limiting formulas as N goes to infinity for the expected distribution of complex zeros of a system of k random fewnomials in m variables. When k = m, for SU(m + 1) polynomials, the limit is the Monge-Ampere measure of a...
Source: http://arxiv.org/abs/1011.3492v1
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Jun 30, 2018
06/18
by
Steve Zelditch
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Let $(\Omega, g)$ be a real analytic Riemannian manifold with real analytic boundary $\partial \Omega$. Let $\psi_{\lambda}$ be an eigenfunction of the Dirichlet-to-Neumann operator $\Lambda$ of $(\Omega, g, \partial \Omega)$ of eigenvalue $\lambda$. Let $\mathcal N_{\lambda_j}$ be its nodal set. Then $\mathcal H^{n-2} (\mathcal N_{\lambda}) \leq C_{g, \Omega} \lambda.$ This proves a conjecture of F. H. Lin and K. Bellova.
Topics: Mathematics, Spectral Theory
Source: http://arxiv.org/abs/1403.0647
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3.0
Jun 30, 2018
06/18
by
Xiaonan Ma; George Marinescu; Steve Zelditch
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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/Szeg\"o kernel asymptotics in the case of a positive line bundle, but no positivity is assumed. We give two proofs, one based on the...
Topics: Complex Variables, Mathematics, Differential Geometry
Source: http://arxiv.org/abs/1406.0201
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Sep 21, 2013
09/13
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Bernard Shiffman; Steve Zelditch
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The Newton polytope $P_f$ of a polynomial $f$ is well known to have a strong impact on its zeros, as in the Kouchnirenko-Bernstein theorem on the number of simultaneous zeros of $m$ polynomials with given Newton polytopes. In this article, we show that $P_f$ also has a strong impact on the distribution of zeros of one or several polynomials. We equip the space of (holomorphic) polynomials of degree $\leq N$ in $m$ complex variables with its usual $SU(m+1)$-invariant Gaussian measure and then...
Source: http://arxiv.org/abs/math/0203074v2
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Jul 20, 2013
07/13
by
Frank Ferrari; Semyon Klevtsov; Steve Zelditch
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The purpose of this article is to propose a new method to define and calculate path integrals over metrics on a K\"ahler manifold. The main idea is to use finite dimensional spaces of Bergman metrics, as an approximation to the full space of K\"ahler 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 K\"ahler metrics. Several examples are considered.
Source: http://arxiv.org/abs/1107.4575v2
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Sep 22, 2013
09/13
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Steve Zelditch
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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...
Source: http://arxiv.org/abs/1210.0834v1
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Sep 22, 2013
09/13
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Bernard Shiffman; Steve Zelditch; Scott Zrebiec
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We give asymptotic large deviations estimates for the volume inside a domain U of the zero set of a random polynomial of degree N, or more generally, of a holomorphic section of the N-th power of a positive line bundle on a compact Kaehler manifold. In particular, we show that for all $\delta>0$, the probability that this volume differs by more than $\delta N$ from its average value is less than $\exp(-C_{\delta,U}N^{m+1})$, for some constant $C_{\delta,U}>0$. As a consequence, the...
Source: http://arxiv.org/abs/0805.2598v2
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Sep 20, 2013
09/13
by
Hans Christianson; John Toth; Steve Zelditch
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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...
Source: http://arxiv.org/abs/1205.0286v2
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Sep 18, 2013
09/13
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Andrew Hassell; Steve Zelditch
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We consider classes of simply connected planar domains which are isophasal, ie, have the same scattering phase $s(\l)$ for all $\l > 0$. This is a scattering-theoretic analogue of isospectral domains. Using the heat invariants and the determinant of the Laplacian, Osgood, Phillips and Sarnak showed that each isospectral class is sequentially compact in a natural $C$-infinity topology. In this paper, we show sequential compactness of each isophasal class of domains. To do this we define the...
Source: http://arxiv.org/abs/math/0002023v1
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Sep 23, 2013
09/13
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Renjie Feng; Steve Zelditch
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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...
Source: http://arxiv.org/abs/1212.4762v1
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Sep 19, 2013
09/13
by
Bernard Shiffman; Steve Zelditch
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We consider ensembles of random polynomials of the form $p(z)=\sum_{j = 1}^N a_j P_j$ where $\{a_j\}$ are independent complex normal random variables and where $\{P_j\}$ are the orthonormal polynomials on the boundary of a bounded simply connected analytic plane domain $\Omega \subset C$ relative to an analytic weight $\rho(z) |dz|$. In the simplest case where $\Omega$ is the unit disk and $\rho=1$, so that $P_j(z) = z^j$, it is known that the average distribution of zeros is the uniform...
Source: http://arxiv.org/abs/math/0206162v1
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Sep 20, 2013
09/13
by
Steve Zelditch
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This short note proves that a Laplacian cannot be quantum uniquely ergodic if it possesses a quasimode of order zero which (i) has a singular limit, and (ii) is a linear combination of a uniformly bounded number of eigenfunctions (modulo an o(1) error). Bouncing ball quasimodes of the stadium are believed to have this property (E.J. Heller et al) and so are analogous quasimodes recently constructed by H. Donnelly on certain non-positively curved surfaces. The main ingredient is the proof that...
Source: http://arxiv.org/abs/math-ph/0301035v2
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Sep 20, 2013
09/13
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Michael R. Douglas; Bernard Shiffman; Steve Zelditch
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A fundamental problem in contemporary string/M theory is to count the number of inequivalent vacua satisfying constraints in a string theory model. This article contains the first rigorous results on the number and distribution of supersymmetric vacua of type IIb string theories compactified on a Calabi-Yau 3-fold $X$ with flux. In particular, complete proofs of the counting formulas in Ashok-Douglas and Denef-Douglas are given, together with van der Corput style remainder estimates. We also...
Source: http://arxiv.org/abs/math-ph/0506015v4
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Sep 18, 2013
09/13
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Steve Zelditch
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This paper concerns spectral invariants of the Laplacian on a compact Riemannian manifold (M,g) known as wave invariants. If U(t) denotes the wave group of (M,g), then the trace Tr U(t) is singular when t = 0 or when ti is the length of a closed geodesic. It has a special type of singularity expansion at each length and the coefficients are known as the wave invariants. Our main purpose is to calculate the wave invariants explicitly in terms of curvature, Jacobi fields etc. when the closed...
Source: http://arxiv.org/abs/math/0002036v1
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Sep 18, 2013
09/13
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Steve Zelditch
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We prove that a two-component mirror-symmetric analytic obstacle in the plane is determined by its resonance poles among such obstacles. The proof is essentially the same as in the interior case (part II of the series). A so-called interior/exterior duality formula is used to simplify the proof. A fair amount of exposition is included for the sake of completeness.
Source: http://arxiv.org/abs/math/0202075v2
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Sep 18, 2013
09/13
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Bernard Shiffman; Tatsuya Tate; Steve Zelditch
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We determine the asymptotics of the joint eigenfunctions of the torus action on a toric Kahler variety. Such varieties are models of completely integrable systems in complex geometry. We first determine the pointwise asymptotics of the eigenfunctions, which show that they behave like Gaussians centered at the corresponding classical torus. We then show that there is a universal Gaussian scaling limit of the distribution function near its center. We also determine the limit distribution for the...
Source: http://arxiv.org/abs/math/0306189v1
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Sep 18, 2013
09/13
by
Steve Zelditch
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This paper concerns the inverse spectral problem for analytic simple surfaces of revolution. By `simple' is meant that there is precisely one critical distance from the axis of revolution. Such surfaces have completely integrable geodesic flows with global action-angle variables and possess global quantum Birkhoff normal forms (Colin de Verdiere). We prove that isospectral surfaces within this class are isometric. The first main step is to show that the normal form at meridian geodesics is a...
Source: http://arxiv.org/abs/math-ph/0002012v1
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Sep 18, 2013
09/13
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Steve Zelditch
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We give a rigorous calculation of the large N limit of the partition function of SU(N) gauge theory on a 2D cylinder in the case where one boundary holomony is a so-called special element of type $\rho$. By MacDonald's identity, the partition function factors in this case as a product over positive roots and it is straightforward to calculate the large N asymptotics of the free energy. We obtain the unexpected result that the free energy in these cases is asymptotic to N times a functional of...
Source: http://arxiv.org/abs/hep-th/0305218v1
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Sep 18, 2013
09/13
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Steve Zelditch; Maciej Zworski
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We prove a scattering theoretical version of the Berry-Tabor conjecture: for an almost every surface in a class of cylindrical surfaces of revolution, the large energy limit of the pair correlation measure of the quantum phase shifts is Poisson, that is, it is given by the uniform measure.
Source: http://arxiv.org/abs/math-ph/0002006v1
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Sep 18, 2013
09/13
by
Pavel Bleher; Bernard Shiffman; Steve Zelditch
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This note is concerned with the scaling limit as N approaches infinity of n-point correlations between zeros of random holomorphic polynomials of degree N in m variables. More generally we study correlations between zeros of holomorphic sections of powers L^N of any positive holomorphic line bundle L over a compact Kahler manifold. Distances are rescaled so that the average density of zeros is independent of N. Our main result is that the scaling limits of the correlation functions and, more...
Source: http://arxiv.org/abs/math-ph/9903012v1
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Sep 19, 2013
09/13
by
Nalini Anantharaman; Steve Zelditch
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We construct an explicit intertwining operator $\lcal$ between the Schr\"odinger group $e^{it \frac\Lap2} $ and the geodesic flow $g^t$ on certain Hilbert spaces of symbols on the cotangent bundle $T^* \X$ of a compact hyperbolic surface $\X = \Gamma \backslash \D$. Thus, the quantization Op(\lcal^{-1} a) satisfies an exact Egorov theorem. The construction of $\lcal$ is based on a complete set of Patterson-Sullivan distributions.
Source: http://arxiv.org/abs/1010.0867v1
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Jul 19, 2013
07/13
by
Bernard Shiffman; Steve Zelditch; Qi Zhong
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We study the conditional distribution of zeros of a Gaussian system of random polynomials (and more generally, holomorphic sections), given that the polynomials or sections vanish at a point p (or a fixed finite set of points). The conditional distribution is analogous to the pair correlation function of zeros, but we show that it has quite a different small distance behavior. In particular, the conditional distribution does not exhibit repulsion of zeros in dimension one. To prove this, we...
Source: http://arxiv.org/abs/1005.4166v1
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Sep 24, 2013
09/13
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Renjie Feng; Steve Zelditch
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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...
Source: http://arxiv.org/abs/1112.3993v1
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Sep 22, 2013
09/13
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Steve Zelditch
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This is a report for the 2003 Forges Les Eaux PDE conference on recent results with A. Hassell on quantum ergodicity of boundary traces of eigenfunctions on domains with ergodic billiards, and of work in progress with Hassell and Sogge on norms of boundary traces. Related work by Burq, Grieser and Smith-Sogge is also discussed.
Source: http://arxiv.org/abs/math/0308220v1
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Sep 21, 2013
09/13
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Steve Zelditch
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This is a survey of the inverse spectral problem on (mainly compact) Riemannian manifolds, with or without boundary. The emphasis is on wave invariants: on how wave invariants have been calculated and how they have been applied to concrete inverse spectral problems.
Source: http://arxiv.org/abs/math/0402356v1
