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7.0

Mar 27, 2016
03/16

Mar 27, 2016
by
Steve Zelditch

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We study the off-diagonal decay of Bergman kernels $\Pi_{h^k}(z,w)$ 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...

Topics: Complex Variables, Mathematics

Source: http://arxiv.org/abs/1603.08281

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2.0

Feb 22, 2016
02/16

Feb 22, 2016
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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6.0

Dec 21, 2015
12/15

Dec 21, 2015
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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2.0

Oct 18, 2015
10/15

Oct 18, 2015
by
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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21

May 20, 2015
05/15

May 20, 2015
by
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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Apr 8, 2015
04/15

Apr 8, 2015
by
Steve Zelditch

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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".

Topics: Spectral Theory, Mathematics

Source: http://arxiv.org/abs/1504.02000

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10.0

Mar 30, 2015
03/15

Mar 30, 2015
by
Renjie Feng; Steve Zelditch

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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...

Topics: Probability, Mathematics

Source: http://arxiv.org/abs/1503.08892

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3.0

Sep 6, 2014
09/14

Sep 6, 2014
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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3.0

Jun 1, 2014
06/14

Jun 1, 2014
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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4.0

Apr 10, 2014
04/14

Apr 10, 2014
by
Steve Zelditch

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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...

Topics: Mathematics, Spectral Theory

Source: http://arxiv.org/abs/1404.2906

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4.0

Apr 2, 2014
04/14

Apr 2, 2014
by
Semyon Klevtsov; Steve Zelditch

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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...

Topics: High Energy Physics - Theory, Mathematics, Mathematical Physics, Differential Geometry

Source: http://arxiv.org/abs/1404.0659

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5.0

Mar 3, 2014
03/14

Mar 3, 2014
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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7.0

Jan 18, 2014
01/14

Jan 18, 2014
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

6
6.0

2014
2014

2014
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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Jun 4, 2013
06/13

Jun 4, 2013
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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47

Mar 17, 2013
03/13

Mar 17, 2013
by
Renjie Feng; Steve Zelditch

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We prove that the expected value and median of the supremum of $L^2$ normalized random holomorphic fields of degree $n$ on $m$-dimensional K\"ahler 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...

Source: http://arxiv.org/abs/1303.4096v1

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98

Jan 13, 2013
01/13

Jan 13, 2013
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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55

Dec 19, 2012
12/12

Dec 19, 2012
by
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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51

Oct 7, 2012
10/12

Oct 7, 2012
by
Steve Zelditch

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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...

Source: http://arxiv.org/abs/1210.2069v1

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51

Oct 2, 2012
10/12

Oct 2, 2012
by
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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58

May 22, 2012
05/12

May 22, 2012
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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45

Apr 23, 2012
04/12

Apr 23, 2012
by
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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58

Dec 16, 2011
12/11

Dec 16, 2011
by
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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68

Dec 15, 2011
12/11

Dec 15, 2011
by
Frank Ferrari; Semyon Klevtsov; Steve Zelditch

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The Mabuchi energy is an interesting geometric functional on the space of K\"ahler metrics that plays a crucial r\^ole in the study of the geometry of K\"ahler 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...

Source: http://arxiv.org/abs/1112.1352v2

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54

Sep 9, 2011
09/11

Sep 9, 2011
by
Christopher D. Sogge; Steve Zelditch

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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, G\'erard 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

Source: http://arxiv.org/abs/1108.2726v2

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45

Jul 30, 2011
07/11

Jul 30, 2011
by
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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167

Jul 3, 2011
07/11

Jul 3, 2011
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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45

Apr 26, 2011
04/11

Apr 26, 2011
by
Hamid Hezari; Steve Zelditch

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We prove that ellipses are infinitesimally spectrally rigid among $C^{\infty}$ domains with the symmetries of the ellipse.

Source: http://arxiv.org/abs/1007.1741v3

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86

Apr 22, 2011
04/11

Apr 22, 2011
by
John Toth; Steve Zelditch

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Quantum ergodic restriction (QER) is the problem of finding conditions on a hypersurface $H$ so that restrictions $\phi_j |_H$ to $H$ of $\Delta$-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 $(\phi_j|H, \partial \phi_j|H)$ is quantum ergodic if the Dirichlet and Neumann data are weighted appropriately. Secondly (ii) we give conditions on $H$ so that...

Source: http://arxiv.org/abs/1005.1636v2

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40

Nov 17, 2010
11/10

Nov 17, 2010
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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71

Nov 15, 2010
11/10

Nov 15, 2010
by
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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37

Oct 31, 2010
10/10

Oct 31, 2010
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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35

Oct 27, 2010
10/10

Oct 27, 2010
by
Yanir A. Rubinstein; Steve Zelditch

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The Cauchy problem for the homogeneous (real and complex) Monge-Ampere equation (HRMA/HCMA) arises from the initial value problem for geodesics in the space of Kahler metrics. It is an ill-posed problem. We conjecture that, in its lifespan, the solution can be obtained by Toeplitz quantizing the Hamiltonian flow defined by the Cauchy data, analytically continuing the quantization, and then taking a kind of logarithmic classical limit. In this article, we prove that in the case of torus...

Source: http://arxiv.org/abs/1008.3577v3

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34

Oct 25, 2010
10/10

Oct 25, 2010
by
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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49

Oct 5, 2010
10/10

Oct 5, 2010
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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44

Sep 27, 2010
09/10

Sep 27, 2010
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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60

May 23, 2010
05/10

May 23, 2010
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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50

Jan 24, 2010
01/10

Jan 24, 2010
by
Hamid Hezari; Steve Zelditch

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In this paper we show that bounded analytic domains in $\R^n$ with mirror symmetries across all coordinate axes are spectrally determined among other such domains. Our approach builds on finding concrete formulas for the wave invariants at a bouncing ball orbit. The wave invariants are calculated from a stationary phase expansion applied to a well-constructed microlocal parametrix for the trace of the resolvent.

Source: http://arxiv.org/abs/0902.1373v3

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42

Aug 5, 2009
08/09

Aug 5, 2009
by
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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98

Mar 19, 2009
03/09

Mar 19, 2009
by
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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47

Mar 2, 2009
03/09

Mar 2, 2009
by
Jian Song; Steve Zelditch

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Geodesics on the infinite dimensional symmetric space $\hcal$ of K\"ahler metrics in a fixed K\"ahler class on a projective K\"ahler manifold X are solutions of a homogeneous complex Monge-Amp\`ere equation in $X \times A$, where $A \subset \C$ is an annulus. They are analogues of 1PS (one-parameter subgroups) on symmetric spaces $G_{\C}/G$. Donaldson, Arezzo-Tian and Phong-Sturm raised the question whether Monge-Amp\`ere geodesics can be approximated by 1PS geodesics in the...

Source: http://arxiv.org/abs/0707.3082v3

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49

Nov 26, 2008
11/08

Nov 26, 2008
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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42

Jun 6, 2008
06/08

Jun 6, 2008
by
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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May 15, 2008
05/08

May 15, 2008
by
Dmitry Jakobson; Alexander Strohmaier; Steve Zelditch

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On a compact K\"ahler manifold there is a canonical action of a Lie-superalgebra on the space of differential forms. It is generated by the differentials, the Lefschetz operator and the adjoints of these operators. We determine the asymptotic distribution of irreducible representations of this Lie-superalgebra on the eigenspaces of the Laplace-Beltrami operator. Because of the high degree of symmetry the Laplace-Beltrami operator on forms can not be quantum ergodic. We show that after...

Source: http://arxiv.org/abs/0805.2376v1

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Mar 30, 2008
03/08

Mar 30, 2008
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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Mar 8, 2008
03/08

Mar 8, 2008
by
Yanir A. Rubinstein; Steve Zelditch

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We generalize the results of Song-Zelditch on geodesics in spaces of Kahler metrics on toric varieties to harmonic maps of any compact Riemannian manifold with boundary into the space of Kahler metrics on a toric variety. We show that the harmonic map equation can always be solved and that such maps may be approximated in the C^2 topology by harmonic maps into the spaces of Bergman metrics. In particular, WZW maps, or equivalently solutions of a homogeneous Monge-Ampere equation on the product...

Source: http://arxiv.org/abs/0803.1249v1

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Jan 22, 2008
01/08

Jan 22, 2008
by
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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Jan 18, 2008
01/08

Jan 18, 2008
by
Steve Zelditch

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This is part of a series of papers on the inverse spectral problem for bounded analytic plane domains. Here, we use the trace formula established in the first paper (`Balian-Bloch trace formula') to explicitly calculate wave trace invariants associated to bouncing ball orbits and dihedral rays. We use these invariants to prove that simply connected bounded analytic plane domains with one symmetry (which reverses a bouncing ball orbit of fixed length L) are spectrally determined in this class.

Source: http://arxiv.org/abs/math/0111078v3

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Dec 21, 2007
12/07

Dec 21, 2007
by
Jian Song; Steve Zelditch

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This article contains a detailed study, in the toric case, of the test configuration geodesic rays defined by Phong-Sturm. We show that the `Bergman approximations' of Phong-Sturm converge in C^1 to the geodesic ray and that the geodesic ray itself is C^{1,1} and no better. The \kahler metrics associated to the geodesic ray of potentials are discontinuous across certain hypersurfaces and are degenerate on certain open sets. A novelty in the analysis is the connection between Bergman metrics,...

Source: http://arxiv.org/abs/0712.3599v1

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Nov 12, 2007
11/07

Nov 12, 2007
by
Bernard Shiffman; Steve Zelditch

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We consider the zero sets $Z_N$ of systems of $m$ random polynomials of degree $N$ in $m$ complex variables, and we give asymptotic formulas for the random variables given by summing a smooth test function over $Z_N$. Our asymptotic formulas show that the variances for these smooth statistics have the growth $N^{m-2}$. We also prove analogues for the integrals of smooth test forms over the subvarieties defined by $k

Source: http://arxiv.org/abs/0711.1840v1