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185
Nov 5, 2007
11/07
Nov 5, 2007
by
John A. Toth; Steve Zelditch
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We consider the zeros on the boundary $\partial \Omega$ of a Neumann eigenfunction $\phi_{\lambda}$ of a real analytic plane domain $\Omega$. We prove that the number of its boundary zeros is $O (\lambda)$ where $-\Delta \phi_{\lambda} = \lambda^2 \phi_{\lambda}$. We also prove that the number of boundary critical points of either a Neumann or Dirichlet eigenfunction is $O(\lambda)$. It follows that the number of nodal lines of $\phi_{\lambda}$ (components of the nodal set) which touch the...
Source: http://arxiv.org/abs/0710.0101v4
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46
Jun 4, 2007
06/07
Jun 4, 2007
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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68
Mar 17, 2007
03/07
Mar 17, 2007
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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53
Feb 10, 2006
02/06
Feb 10, 2006
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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50
Dec 10, 2005
12/05
Dec 10, 2005
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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53
Nov 18, 2005
11/05
Nov 18, 2005
by
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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53
Aug 8, 2005
08/05
Aug 8, 2005
by
Steve Zelditch
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We determine the limit distribution (as $\lambda \to \infty$) of complex zeros for holomorphic continuations $\phi_{\lambda}^{\C}$ to Grauert tubes of real eigenfunctions of the Laplacian on a real analytic compact Riemannian manifold $(M, g)$ with ergodic geodesic flow. If $\{\phi_{j_k} \}$ is an ergodic sequence of eigenfunctions, we prove the weak limit formula $\frac{1}{\lambda_j} [Z_{\phi_{j_k}^{\C}}] \to \frac{i}{\pi} \bar{\partial} {\partial} |\xi|_g$, where $ [Z_{\phi_{j_k}^{\C}}]$ is...
Source: http://arxiv.org/abs/math/0505513v4
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61
Sep 23, 2004
09/04
Sep 23, 2004
by
Jens Marklof; Stephen O'Keefe; Steve Zelditch
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For a general class of unitary quantum maps, whose underlying classical phase space is divided into several invariant domains of positive measure, we establish analogues of Weyl's law for the distribution of eigenphases. If the map has one ergodic component, and is periodic on the remaining domains, we prove the Schnirelman-Zelditch-Colin de Verdiere Theorem on the equidistribution of eigenfunctions with respect to the ergodic component of the classical map (quantum ergodicity). We apply our...
Source: http://arxiv.org/abs/nlin/0404038v2
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56
Jun 4, 2004
06/04
Jun 4, 2004
by
Michael R. Douglas; Bernard Shiffman; Steve Zelditch
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Supersymmetric vacua (`universes') of string/M theory may be identified with certain critical points of a holomorphic section (the `superpotential') of a Hermitian holomorphic line bundle over a complex manifold. An important physical problem is to determine how many vacua there are and how they are distributed. The present paper initiates the study of the statistics of critical points $\nabla s = 0$ of Gaussian random holomorphic sections with respect to a connection $\nabla$. Even the...
Source: http://arxiv.org/abs/math/0402326v2
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61
Feb 23, 2004
02/04
Feb 23, 2004
by
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
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49
Aug 23, 2003
08/03
Aug 23, 2003
by
Steve Zelditch
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What does it mean to quantize a symplectic map $\chi$? In deformation quantization, it means to construct an automorphism of the $*$ algebra associated to $\chi$. In quantum chaos it means to construct unitary operators $U_{\chi}$ such that $A \to U_{\chi} A U_{\chi}^*$ defines an automorphism of the algebra of observables. In geometric quantization and in PDE it means to construct a unitary Fourier integral (or Toeplitz) operator associated to the graph of $\chi$. We compare the definitions in...
Source: http://arxiv.org/abs/math/0307175v2
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54
Aug 23, 2003
08/03
Aug 23, 2003
by
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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52
Jul 31, 2003
07/03
Jul 31, 2003
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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59
Jun 20, 2003
06/03
Jun 20, 2003
by
Steve Zelditch
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We review some recent results on asymptotic properties of polynomials of large degree, of general holomorphic sections of high powers of positive line bundles over Kahler manifolds, and of Laplace eigenfunctions of large eigenvalue on compact Riemannian manifolds. We describe statistical patterns in the zeros, critical points and L^p norms of random polynomials and holomorphic sections, and the influence of the Newton polytope on these patterns. For eigenfunctions, we discuss L^p norms and mass...
Source: http://arxiv.org/abs/math/0208104v2
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39
Jun 11, 2003
06/03
Jun 11, 2003
by
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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50
May 24, 2003
05/03
May 24, 2003
by
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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81
May 17, 2003
05/03
May 17, 2003
by
Tatsuya Tate; Steve Zelditch
texts
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We give asymptotic formulas for the multiplicities of weights and irreducible summands in high-tensor powers $V_{\lambda}^{\otimes N}$ of an irreducible representation $V_{\lambda}$ of a compact connected Lie group $G$. The weights are allowed to depend on $N$, and we obtain several regimes of pointwise asymptotics, ranging from a central limit region to a large deviations region. We use a complex steepest descent method that applies to general asymptotic counting problems for lattice paths...
Source: http://arxiv.org/abs/math/0305251v1
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81
Apr 8, 2003
04/03
Apr 8, 2003
by
Steve Zelditch
texts
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We give a rigorous version of the classical Balian-Bloch trace formula, a semiclassical expansion around a periodic reflecting ray of the (regularized) resolvent of the Dirichlet Laplacian on a bounded smooth plane domain. It is equivalent to the Poisson relation (or wave trace formula) between spectrum and closed geodesics. We view it primarily as a computational device for explicitly calculating wave trace invariants. Its effectiveness will be illustrated in subsquent articles in the series...
Source: http://arxiv.org/abs/math/0111077v3
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41
Aug 6, 2002
08/02
Aug 6, 2002
by
John A. Toth; Steve Zelditch
texts
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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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59
Jun 17, 2002
06/02
Jun 17, 2002
by
Bernard Shiffman; Steve Zelditch
texts
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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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50
Mar 12, 2002
03/02
Mar 12, 2002
by
Bernard Shiffman; Steve Zelditch
texts
eye 50
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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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58
Feb 8, 2002
02/02
Feb 8, 2002
by
John A. Toth; Steve Zelditch
texts
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This is a partly expository, partly new paper on sup norm estimates of eigenfunctions. The focus is on the quantum completely integrable case. We give a new proof of the main result of our paper ``Riemannian manifolds with uniformly bounded eigenfunctions' (Duke Math J. 111 (2002), 97-132), based on the analysis of quasi-modes and Birkhoff normal forms. We also discuss related issues of resonant normal forms and tunnelling between tori.
Source: http://arxiv.org/abs/math/0202074v1
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52
May 3, 2001
05/01
May 3, 2001
by
Christopher D. Sogge; Steve Zelditch
texts
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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/math/0103172v2
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55
Feb 5, 2001
02/01
Feb 5, 2001
by
John Toth; Steve Zelditch
texts
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The standard eigenfunctions $\phi_{\lambda} = e^{i < \lambda, x >}$ on flat tori $\R^n / L$ have $L^{\infty}$-norms bounded independently of the eigenvalue. In the case of irrational flat tori, it follows that $L^2$-normalized eigenfunctions have uniformly bounded $L^{\infty}$-norms. Similar bases exist on other flat manifolds. Does this property characterize flat manifolds? We give an affirmative answer for compact Riemannian manifolds with completely integrable geodesic flows.
Source: http://arxiv.org/abs/math-ph/0002038v2
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38
Nov 20, 2000
11/00
Nov 20, 2000
by
Pavel Bleher; Bernard Shiffman; Steve Zelditch
texts
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In our previous work [math-ph/9904020], we proved that the correlation functions for simultaneous zeros of random generalized polynomials have universal scaling limits and we gave explicit formulas for pair correlations in codimensions 1 and 2. The purpose of this paper is to compute these universal limits in all dimensions and codimensions. First, we use a supersymmetry method to express the n-point correlations as Berezin integrals. Then we use the Wick method to give a closed formula for the...
Source: http://arxiv.org/abs/math-ph/0011016v2
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111
Oct 17, 2000
10/00
Oct 17, 2000
by
Pavel Bleher; Bernard Shiffman; Steve Zelditch
texts
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This article is concerned with random holomorphic polynomials and their generalizations to algebraic and symplectic geometry. A natural algebro-geometric generalization studied in our prior work involves random holomorphic sections $H^0(M,L^N)$ of the powers of any positive line bundle $L \to M$ over any complex manifold. Our main interest is in the statistics of zeros of $k$ independent sections (generalized polynomials) of degree $N$ as $N\to\infty$. We fix a point $P$ and focus on the ball...
Source: http://arxiv.org/abs/math-ph/0002039v2
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55
Oct 10, 2000
10/00
Oct 10, 2000
by
Steve Zelditch
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We review some recent results on random polynomials and their generalizations in complex and symplectic geometry. The main theme is the universality of statistics of zeros and critical points of (generalized) polynomials of degree $N$ on length scales of order $\frac{D}{\sqrt{N}}$ (complex case), resp. $\frac{D}{N}$ (real case).
Source: http://arxiv.org/abs/math-ph/0010012v1
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70
Feb 5, 2000
02/00
Feb 5, 2000
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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54
Feb 4, 2000
02/00
Feb 4, 2000
by
Steve Zelditch
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This paper generalizes the methods and results of our article xxx.lanl.gov math.SP/0002036 from elliptic to general non-degenerate closed geodesics. The main purpose is to introduce a quantum Birkhoff normal form of the Laplacian at a general non-degenerate closed geodesic in the sense of V.Guillemin. Guillemin proved that the coefficients of the normal form at an elliptic closed geodesic could be determined from the wave invariants of this geodesic. We give a new proof, and extend its range to...
Source: http://arxiv.org/abs/math/0002037v1
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53
Feb 4, 2000
02/00
Feb 4, 2000
by
Steve Zelditch
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In this addendum we strengthen the results of math-ph/0002010 in the case of polynomial phases. We prove that Cesaro means of the pair correlation functions of certain integrable quantum maps on the 2-sphere at level N tend almost always to the Poisson (uniform limit). The quantum maps are exponentials of Hamiltonians which have the form a p(I) + b I, where I is the action, where p is a polynomial and where a,b are two real numbers. We prove that for any such family and for almost all a,b, the...
Source: http://arxiv.org/abs/math-ph/0002011v1
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53
Feb 4, 2000
02/00
Feb 4, 2000
by
Steve Zelditch
texts
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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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49
Feb 4, 2000
02/00
Feb 4, 2000
by
Steve Zelditch
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We consider the eigenvalue pair correlation problem for certain integrable quantum maps on the 2-sphere. The classical maps are time one maps of Hamiltonian flows of perfect Morse functions. The quantizations are unitary operators on spaces of homogeneous holomorphic polynomials of degree N in two complex variables. There are N eigenphases on the unit circle and the pair correlation problem is to determine the distribution of spacings between the eigenphases on the length scale of the mean...
Source: http://arxiv.org/abs/math-ph/0002010v1
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66
Feb 3, 2000
02/00
Feb 3, 2000
by
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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48
Feb 3, 2000
02/00
Feb 3, 2000
by
Steve Zelditch
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Quantized contact transformations are Toeplitz operators over a contact manifold $(X,\alpha)$ of the form $U_{\chi} = \Pi A \chi \Pi$, where $\Pi : H^2(X) \to L^2(X)$ is a Szego projector, where $\chi$ is a contact transformation and where $A$ is a pseudodifferential operator over $X$. They provide a flexible alternative to the Kahler quantization of symplectic maps, and encompass many of the examples in the physics literature, e.g. quantized cat maps and kicked rotors. The index problem is to...
Source: http://arxiv.org/abs/math-ph/0002007v1
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47
Feb 3, 2000
02/00
Feb 3, 2000
by
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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44
Feb 3, 2000
02/00
Feb 3, 2000
by
Steve Zelditch
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This paper contains a very simple and general proof that eigenfunctions of quantizations of classically ergodic systems become uniformly distributed in phase space. This ergodicity property of eigenfunctions f is shown to follow from a convexity inequality for the invariant states (Af,f). This proof of ergodicity of eigenfunctions simplifies previous proofs (due to A.I. Shnirelman, Colin de Verdiere and the author) and extends the result to the much more general framework of C* dynamical...
Source: http://arxiv.org/abs/math-ph/0002008v1
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46
Feb 3, 2000
02/00
Feb 3, 2000
by
Steve Zelditch; Maciej Zworski
texts
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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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75
Feb 2, 2000
02/00
Feb 2, 2000
by
Bernard Shiffman; Steve Zelditch
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The spaces $H^0(M, L^N)$ of holomorphic sections of the powers of an ample line bundle $L$ over a compact K\"ahler manifold $(M,\omega)$ have been generalized by Boutet de Monvel and Guillemin to spaces $H^0_J(M, L^N)$ of `almost holomorphic sections' of ample line bundles over an almost complex symplectic manifold $(M, J, \omega)$. We consider the unit spheres $SH^0_J(M, L^N)$ in the spaces $H^0_J(M, L^N)$, which we equip with natural inner products. Our purpose is to show that, in a...
Source: http://arxiv.org/abs/math/0001102v2
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64
Apr 21, 1999
04/99
Apr 21, 1999
by
Pavel Bleher; Bernard Shiffman; Steve Zelditch
texts
eye 64
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We study the limit as $N\to\infty$ of the correlations between simultaneous zeros of random sections of the powers $L^N$ of a positive holomorphic line bundle $L$ over a compact complex manifold $M$, when distances are rescaled so that the average density of zeros is independent of $N$. We show that the limit correlation is independent of the line bundle and depends only on the dimension of $M$ and the codimension of the zero sets. We also provide some explicit formulas for pair correlations....
Source: http://arxiv.org/abs/math-ph/9904020v1
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59
Apr 7, 1999
04/99
Apr 7, 1999
by
Steve Zelditch
texts
eye 59
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Let ${\cal D}$ denote the class of bounded real analytic plane domains with the symmetry of an ellipse. We prove that if $\Omega_1, \Omega_2 \in {\cal D}$ and if the Dirichlet spectra coincide, $Spec(\Omega_1) = Spec(\Omega_2)$, then $\Omega_1 = \Omega_2$ up to rigid motion.
Source: http://arxiv.org/abs/math/9901005v2
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67
Mar 5, 1999
03/99
Mar 5, 1999
by
Pavel Bleher; Bernard Shiffman; Steve Zelditch
texts
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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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49
Mar 12, 1998
03/98
Mar 12, 1998
by
Bernard Shiffman; Steve Zelditch
texts
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We study the limit distribution of zeros of certain sequences of holomorphic sections of high powers $L^N$ of a positive holomorphic Hermitian line bundle $L$ over a compact complex manifold $M$. Our first result concerns `random' sequences of sections. Using the natural probability measure on the space of sequences of orthonormal bases $\{S^N_j\}$ of $H^0(M, L^N)$, we show that for almost every sequence $\{S^N_j\}$, the associated sequence of zero currents $1/N Z_{S^N_j}$ tends to the...
Source: http://arxiv.org/abs/math/9803052v1
