arXiv subjects

Steve Zelditch

Publications and source records attributed to Steve Zelditch.

138 records · Page 8Linked to original sources

arXiv2000

Level Spacings for Integrable Quantum Maps in Genus Zero

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 level spacing. The Berry-Tabor conjecture says that for generic integrable systems, the spacings distribution should tend to the Poisson (uniform) limit as N tends to infinity. In this paper and in its addendum we prove a somewhat weak form of this conjecture: We define a large class of two parameter families of integrable Hamiltonians, and prove that for almost all elements of each family the limit pair correlation function is Poisson. In this paper, we take the limit along a slightly sparse subsequence of N's; in the addendum we take Cesaro means along the full seuqence.

math-ph↗

arXiv2000

Addendum: Level Spacings for Integrable Quantum Maps in Genus Zero

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 pair correlation tends to Poisson on average in N. The results involve Weyl estimates on exponential sums and new metric results on continued fractions. They were motivated by a comparison of the results of math-ph/0002010 with some independent results on pair correlation of fractional parts of polynomials by Rudnick-Sarnak.

math-ph↗

arXiv2000

Index and Dynamics of Quantized Contact Transformations

Quantized contact transformations are Toeplitz operators over a contact manifold $(X,α)$ of the form $U_χ = ΠA χΠ$, where $Π: H^2(X) \to L^2(X)$ is a Szego projector, where $χ$ 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 determine $ind(U_χ)$ when the principal symbol is unitary, or equivalently to determine whether $A$ can be chosen so that $U_χ$ is unitary. We show that the answer is yes in the case of quantized symplectic torus automorphisms $g$---by showing that $U_g$ duplicates the classical transformation laws on theta functions. Using the Cauchy-Szego kernel on the Heisenberg group, we calculate the traces on theta functions of each degree N. We also study the quantum dynamics generated by a general q.c.t. $U_χ$, i.e. the quasi-classical asymptotics of the eigenvalues and eigenfunctions under various ergodicity and mixing hypotheses on $χ.$ Our principal results are proofs of equidistribution of eigenfunctions $ϕ_{Nj}$ and weak mixing properties of matrix elements $(Bϕ_{Ni}, ϕ_{Nj})$ for quantizations of mixing symplectic maps.

math-ph↗

arXiv2000

Szego kernels and a theorem of Tian

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.

math-ph↗

arXiv2000

Determinants of Laplacians in Exterior Domains

We consider classes of simply connected planar domains which are isophasal, ie, have the same scattering phase $s(ł)$ for all $ł> 0$. 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 determinant of the exterior Laplacian and use it together with the heat invariants (the heat invariants and the determinant being isophasal invariants). We show that the determinant of the interior and exterior Laplacians satisfy a Burghelea-Friedlander-Kappeler type surgery formula. This allows a reduction to a problem on bounded domains for which the methods of Osgood, Phillips and Sarnak can be adapted.

math.AP↗

arXiv2000

Quantum ergodicity of C* dynamical systems

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 systems.

math-ph↗

arXiv2000

Spacings between phase shifts in a simple scattering problem

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.

math-ph↗

arXiv2000

Random almost holomorphic sections of ample line bundles on symplectic manifolds

The spaces $H^0(M, L^N)$ of holomorphic sections of the powers of an ample line bundle $L$ over a compact Kähler manifold $(M,ω)$ 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 $(M, J, ω)$. 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 probabilistic sense, almost holomorphic sections behave like holomorphic sections as $N \to \infty$. Our first main result is that almost all sequences of sections $s_N \in SH^0_J(M, L^N)$ are `asymptotically holomorphic' in the Donaldson-Auroux sense that $||s_N||_{\infty}/||s_N||_{2} = O(\sqrt{\log N})$, $||\bar{\partial} s_N||_{\infty}/||s_N||_{2} = O(\sqrt{\log N})$ and $||\partial s_N||_{\infty}/||s_N||_{2} = O(\sqrt{N \log N})$. Our second main result concerns the joint probability distribution of the random variables $s_N(z^p),\ \nabla s_N(z^p)$, $1\le p\le n$, for $n$ distinct points $z^1,..., z^n$ in a neighborhood of a point $P_0\in M$. We show that this joint distribution has a universal scaling limit about $P_0$ as $N \to \infty$. In particular, the limit is precisely the same as in the complex holomorphic case. Our methods involve near-diagonal scaling asymptotics of the Szegö projector $Π_N$ onto $H^0_J(M, L^N)$, which also yields proofs of symplectic analogues of the Kodaira embedding theorem and Tian asymptotic isometry theorem.

math.SG↗

arXiv1999

Universality and scaling of correlations between zeros on complex manifolds

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. In particular, we provide an alternate derivation of Hannay's limit pair correlation function for SU(2) polynomials, and we show that this correlation function holds for all compact Riemann surfaces.

math-ph↗

arXiv1999

Spectral determination of analytic axi-symmetric plane domains

Let ${\cal D}$ denote the class of bounded real analytic plane domains with the symmetry of an ellipse. We prove that if $Ω_1, Ω_2 \in {\cal D}$ and if the Dirichlet spectra coincide, $Spec(Ω_1) = Spec(Ω_2)$, then $Ω_1 = Ω_2$ up to rigid motion.

math.SP↗

arXiv1999

Poincare-Lelong approach to universality and scaling of correlations between zeros

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 generally, of the "correlation forms" are universal, i.e. independent of the bundle L, manifold M or point on M.

math-ph↗

arXiv1998

Distribution of zeros of random and quantum chaotic sections of positive line bundles

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 curvature form $ω$ of $L$. Thus, the zeros of a sequence of sections $s_N \in H^0(M, L^N)$ chosen independently and at random become uniformly distributed. Our second result concerns the zeros of quantum ergodic eigenfunctions, where the relevant orthonormal bases $\{S^N_j\}$ of $H^0(M, L^N)$ consist of eigensections of a quantum ergodic map. We show that also in this case the zeros become uniformly distributed.

math.CV↗