49
49
Sep 22, 2013
09/13
by
Steve Zelditch
texts
eye 49
favorite 0
comment 0
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
5
5.0
Jun 30, 2018
06/18
by
Steve Zelditch
texts
eye 5
favorite 0
comment 0
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
9
9.0
Jun 29, 2018
06/18
by
Steve Zelditch
texts
eye 9
favorite 0
comment 0
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
71
71
Sep 18, 2013
09/13
by
Steve Zelditch
texts
eye 71
favorite 0
comment 0
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
49
49
Sep 18, 2013
09/13
by
Steve Zelditch
texts
eye 49
favorite 0
comment 0
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
100
100
Sep 23, 2013
09/13
by
Steve Zelditch
texts
eye 100
favorite 0
comment 0
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
169
169
Jul 20, 2013
07/13
by
Steve Zelditch
texts
eye 169
favorite 0
comment 0
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
67
67
Sep 18, 2013
09/13
by
Steve Zelditch
texts
eye 67
favorite 0
comment 0
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
51
51
Sep 22, 2013
09/13
by
Steve Zelditch
texts
eye 51
favorite 0
comment 0
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
51
51
Sep 18, 2013
09/13
by
Steve Zelditch
texts
eye 51
favorite 0
comment 0
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
60
60
Sep 18, 2013
09/13
by
Steve Zelditch
texts
eye 60
favorite 0
comment 0
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
50
50
Sep 18, 2013
09/13
by
Steve Zelditch
texts
eye 50
favorite 0
comment 0
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
60
60
Sep 19, 2013
09/13
by
Steve Zelditch
texts
eye 60
favorite 0
comment 0
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
56
56
Sep 19, 2013
09/13
by
Steve Zelditch
texts
eye 56
favorite 0
comment 0
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
6
6.0
Jun 30, 2018
06/18
by
Steve Zelditch
texts
eye 6
favorite 0
comment 0
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
81
81
Sep 22, 2013
09/13
by
Steve Zelditch
texts
eye 81
favorite 0
comment 0
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
3
3.0
Jun 28, 2018
06/18
by
Steve Zelditch
texts
eye 3
favorite 0
comment 0
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
58
58
Sep 18, 2013
09/13
by
Steve Zelditch
texts
eye 58
favorite 0
comment 0
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
44
44
Sep 18, 2013
09/13
by
Steve Zelditch
texts
eye 44
favorite 0
comment 0
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
58
58
Sep 18, 2013
09/13
by
Steve Zelditch
texts
eye 58
favorite 0
comment 0
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
52
52
Sep 20, 2013
09/13
by
Steve Zelditch
texts
eye 52
favorite 0
comment 0
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
53
53
Sep 18, 2013
09/13
by
Steve Zelditch
texts
eye 53
favorite 0
comment 0
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
53
53
Sep 22, 2013
09/13
by
Steve Zelditch
texts
eye 53
favorite 0
comment 0
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
55
55
Sep 22, 2013
09/13
by
Steve Zelditch
texts
eye 55
favorite 0
comment 0
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
19
19
Jun 27, 2018
06/18
by
Steve Zelditch
texts
eye 19
favorite 0
comment 0
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
47
47
Sep 18, 2013
09/13
by
Steve Zelditch
texts
eye 47
favorite 0
comment 0
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
54
54
Sep 18, 2013
09/13
by
Steve Zelditch
texts
eye 54
favorite 0
comment 0
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
53
53
Sep 22, 2013
09/13
by
Steve Zelditch
texts
eye 53
favorite 0
comment 0
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
55
55
Sep 18, 2013
09/13
by
Steve Zelditch
texts
eye 55
favorite 0
comment 0
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
54
54
Sep 18, 2013
09/13
by
Steve Zelditch
texts
eye 54
favorite 0
comment 0
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
62
62
Sep 21, 2013
09/13
by
Steve Zelditch
texts
eye 62
favorite 0
comment 0
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
5
5.0
Jun 30, 2018
06/18
by
Semyon Klevtsov; Steve Zelditch
texts
eye 5
favorite 0
comment 0
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
4
4.0
Jun 30, 2018
06/18
by
Chris Sogge; Steve Zelditch
texts
eye 4
favorite 0
comment 0
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
77
77
Sep 22, 2013
09/13
by
Bernard Shiffman; Steve Zelditch
texts
eye 77
favorite 0
comment 0
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
52
52
Sep 20, 2013
09/13
by
Bernard Shiffman; Steve Zelditch
texts
eye 52
favorite 0
comment 0
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
13
13
Jun 27, 2018
06/18
by
Renjie Feng; Steve Zelditch
texts
eye 13
favorite 0
comment 0
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
24
24
Jun 27, 2018
06/18
by
Semyon Klevtsov; Steve Zelditch
texts
eye 24
favorite 0
comment 0
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
48
48
Sep 18, 2013
09/13
by
Andrew Hassell; Steve Zelditch
texts
eye 48
favorite 0
comment 0
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
46
46
Jul 20, 2013
07/13
by
Hamid Hezari; Steve Zelditch
texts
eye 46
favorite 0
comment 0
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
46
46
Sep 17, 2013
09/13
by
Bernard Shiffman; Steve Zelditch
texts
eye 46
favorite 0
comment 0
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
47
47
Sep 18, 2013
09/13
by
Steve Zelditch; Maciej Zworski
texts
eye 47
favorite 0
comment 0
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
48
48
Sep 18, 2013
09/13
by
Jian Song; Steve Zelditch
texts
eye 48
favorite 0
comment 0
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
54
54
Sep 20, 2013
09/13
by
Nalini Anantharaman; Steve Zelditch
texts
eye 54
favorite 0
comment 0
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
53
53
Sep 21, 2013
09/13
by
Jian Song; Steve Zelditch
texts
eye 53
favorite 0
comment 0
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
59
59
Sep 24, 2013
09/13
by
Renjie Feng; Steve Zelditch
texts
eye 59
favorite 0
comment 0
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
45
45
Sep 19, 2013
09/13
by
Renjie Feng; Steve Zelditch
texts
eye 45
favorite 0
comment 0
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
73
73
Sep 21, 2013
09/13
by
Bernard Shiffman; Steve Zelditch
texts
eye 73
favorite 0
comment 0
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
51
51
Sep 21, 2013
09/13
by
Hamid Hezari; Steve Zelditch
texts
eye 51
favorite 0
comment 0
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
7
7.0
Jun 28, 2018
06/18
by
Yannick Bonthonneau; Steve Zelditch
texts
eye 7
favorite 0
comment 0
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
58
58
Sep 23, 2013
09/13
by
Renjie Feng; Steve Zelditch
texts
eye 58
favorite 0
comment 0
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
