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4.0

Jun 30, 2018
06/18

Jun 30, 2018
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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4.0

Jun 30, 2018
06/18

Jun 30, 2018
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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5.0

Jun 30, 2018
06/18

Jun 30, 2018
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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5.0

Jun 30, 2018
06/18

Jun 30, 2018
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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6.0

Jun 30, 2018
06/18

Jun 30, 2018
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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Jun 30, 2018
06/18

Jun 30, 2018
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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7.0

Jun 30, 2018
06/18

Jun 30, 2018
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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9.0

Jun 29, 2018
06/18

Jun 29, 2018
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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3.0

Jun 29, 2018
06/18

Jun 29, 2018
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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7.0

Jun 28, 2018
06/18

Jun 28, 2018
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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4.0

Jun 28, 2018
06/18

Jun 28, 2018
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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Jun 27, 2018
06/18

Jun 27, 2018
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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Jun 27, 2018
06/18

Jun 27, 2018
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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Jun 27, 2018
06/18

Jun 27, 2018
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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55

Sep 24, 2013
09/13

Sep 24, 2013
by
Christopher D. Sogge; 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/math/0103172v2

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60

Sep 24, 2013
09/13

Sep 24, 2013
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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46

Sep 24, 2013
09/13

Sep 24, 2013
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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71

Sep 23, 2013
09/13

Sep 23, 2013
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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68

Sep 23, 2013
09/13

Sep 23, 2013
by
Pavel Bleher; Bernard Shiffman; Steve Zelditch

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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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Sep 23, 2013
09/13

Sep 23, 2013
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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100

Sep 23, 2013
09/13

Sep 23, 2013
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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48

Sep 23, 2013
09/13

Sep 23, 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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58

Sep 23, 2013
09/13

Sep 23, 2013
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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49

Sep 22, 2013
09/13

Sep 22, 2013
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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77

Sep 22, 2013
09/13

Sep 22, 2013
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

Sep 22, 2013
09/13

Sep 22, 2013
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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45

Sep 22, 2013
09/13

Sep 22, 2013
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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55

Sep 22, 2013
09/13

Sep 22, 2013
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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53

Sep 22, 2013
09/13

Sep 22, 2013
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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56

Sep 22, 2013
09/13

Sep 22, 2013
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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45

Sep 22, 2013
09/13

Sep 22, 2013
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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52

Sep 22, 2013
09/13

Sep 22, 2013
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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81

Sep 22, 2013
09/13

Sep 22, 2013
by
Steve Zelditch

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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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39

Sep 22, 2013
09/13

Sep 22, 2013
by
Pavel Bleher; Bernard Shiffman; Steve Zelditch

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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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51

Sep 21, 2013
09/13

Sep 21, 2013
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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59

Sep 21, 2013
09/13

Sep 21, 2013
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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62

Sep 21, 2013
09/13

Sep 21, 2013
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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55

Sep 21, 2013
09/13

Sep 21, 2013
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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53

Sep 21, 2013
09/13

Sep 21, 2013
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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51

Sep 21, 2013
09/13

Sep 21, 2013
by
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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Sep 21, 2013
09/13

Sep 21, 2013
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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Sep 21, 2013
09/13

Sep 21, 2013
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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Sep 21, 2013
09/13

Sep 21, 2013
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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Sep 20, 2013
09/13

Sep 20, 2013
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

Sep 20, 2013
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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Sep 20, 2013
09/13

Sep 20, 2013
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 20, 2013
09/13

Sep 20, 2013
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 20, 2013
09/13

Sep 20, 2013
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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Sep 20, 2013
09/13

Sep 20, 2013
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 20, 2013
09/13

Sep 20, 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