arXiv subjects

Steve Zelditch

Publications and source records attributed to Steve Zelditch.

At least 91 records · Page 5Linked to original sources

arXiv2010

Random complex fewnomials, I

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 toric Kaehler potential on CP^m obtained by averaging a `discrete Legendre transform' of the Fubini-Study symplectic potential at f points of the unit simplex in R^m.

math.CV↗

arXiv2010

Concerning the $L^4$ norms of typical eigenfunctions on compact surfaces

Let $(M,g)$ be a two-dimensional compact boundaryless Riemannian manifold with Laplacian, $Δ_g$. If $e_λ$ are the associated eigenfunctions of $\sqrt{-Δ_g}$ so that $-Δ_g e_λ= λ^2 e_λ$, then it has been known for some time \cite{soggeest} that $\|e_λ\|_{L^4(M)}\lesssim λ^{1/8}$, assuming that $e_λ$ 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 weight spherical harmonics of degree $k$. On the other hand, we shall show that the average $L^4$ norm of the standard basis for the space ${\mathcal H}_k$ of spherical harmonics of degree $k$ on $S^2$ merely grows like $(\log k)^{1/4}$. We also sketch a proof that the average of $\sum_{j = 1}^{2k + 1} \|e_λ\|_{L^4}^4$ for a random orthonormal basis of ${\mathcal H}_k$ is O(1). We are not able to determine the maximum of this quantity over all orthonormal bases of ${\mathcal H}_k$ or for orthonormal bases of eigenfunctions on other Riemannian manifolds. However, under the assumption that the periodic geodesics in $(M,g)$ are of measure zero, we are able to show that for {\it any} orthonormal basis of eigenfunctions we have that $\|e_{λ_{j_k}}\|_{L^4(M)}=o(λ_{j_k}^{1/8})$ for a density one subsequence of eigenvalues $λ_{j_k}$. This assumption is generic and it is the one in the Duistermaat-Gullemin theorem \cite{dg} which gave related improvements for the error term in the sharp Weyl theorem. The proof of our result uses a recent estimate of the first author \cite{Sokakeya} that gives a necessary and sufficient condition that $\|e_λ\|_{L^4(M)}=o(λ^{1/8})$.

math.AP↗

arXiv2010

The Cauchy problem for the homogeneous Monge-Ampere equation, I. Toeplitz quantization

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 invariant metrics (where the HCMA reduces to the HRMA) this "quantum analytic continuation potential" coincides with the well-known Legendre transform potential, and hence solves the equation as long as it is smooth. In the sequel we prove that the Legendre transform potential ceases to solve the HRMA after that time.

math.DG↗

arXiv2010

The Cauchy problem for the homogeneous Monge-Ampere equation, II. Legendre transform

We continue our study of the Cauchy problem for the homogeneous (real and complex) Monge-Ampere equation (HRMA/HCMA). In the prequel a quantum mechanical approach for solving the HCMA was developed, and was shown to coincide with the well-known Legendre transform approach in the case of the HRMA. In this article---that uses tools of convex analysis and can be read independently---we prove that the candidate solution produced by these methods ceases to solve the HRMA, even in a weak sense, as soon as it ceases to be differentiable.

math.AP↗

arXiv2010

Intertwining the geodesic flow and the Schrodinger group on hyperbolic surfaces

We construct an explicit intertwining operator $\lcal$ between the Schrödinger group $e^{it \frac\Lap2} $ and the geodesic flow $g^t$ on certain Hilbert spaces of symbols on the cotangent bundle $T^* \X$ of a compact hyperbolic surface $\X = Γ\backslash \D$. Thus, the quantization Op(\lcal^{-1} a) satisfies an exact Egorov theorem. The construction of $\lcal$ is based on a complete set of Patterson-Sullivan distributions.

math.SP↗

arXiv2010

Large deviations for zeros of $P(ϕ)_2$ random polynomials

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 $P(ϕ)_2$ 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 $P(ϕ)_2$ 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 distribution of zeros in the $P(ϕ)_2$ ensembles tends to the same equilibrium measure as in the Gaussian case.

math.PR↗

arXiv2010

Random zeros on complex manifolds: conditional expectations

We study the conditional distribution of zeros of a Gaussian system of random polynomials (and more generally, holomorphic sections), given that the polynomials or sections vanish at a point p (or a fixed finite set of points). The conditional distribution is analogous to the pair correlation function of zeros, but we show that it has quite a different small distance behavior. In particular, the conditional distribution does not exhibit repulsion of zeros in dimension one. To prove this, we give universal scaling asymptotics for the conditional zero distribution around p. The key tool is the conditional Szego kernel and its scaling asymptotics.

math.CV↗

arXiv2010

Inverse spectral problem for analytic $(Z/2Z)^n$-symmetric domains in $R^n$

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.

math.AP↗

arXiv2009

About the blowup of quasimodes on Riemannian manifolds

On any compact Riemannian manifold $(M, g)$ of dimension $n$, the $L^2$-normalized eigenfunctions ${ϕ_λ}$ satisfy $||ϕ_λ||_{\infty} \leq C λ^{\frac{n-1}{2}}$ where $-Δϕ_λ = λ^2 ϕ_λ.$ 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 $\R^n/Γ$. We say that $S^n$, but not $\R^n/Γ$, is a Riemannian manifold with maximal eigenfunction growth. The problem which motivates this paper is to determine the $(M, g)$ with maximal eigenfunction growth. In an earlier work, two of us showed that such an $(M, g)$ must have a point $x$ where the set ${\mathcal L}_x$ of geodesic loops at $x$ has positive measure in $S^*_x M$. We strengthen this result here by showing that such a manifold must have a point where the set ${\mathcal R}_x$ of recurrent directions for the geodesic flow through x satisfies $|{\mathcal R}_x|>0$. We also show that if there are no such points, $L^2$-normalized quasimodes have sup-norms that are $o(λ^{n-1)/2})$, and, in the other extreme, we show that if there is a point blow-down $x$ at which the first return map for the flow is the identity, then there is a sequence of quasi-modes with $L^\infty$-norms that are $Ω(λ^{(n-1)/2})$.

math.AP↗

arXiv2009

Local and global analysis of eigenfunctions

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.

math.AP↗

arXiv2009

Bergman metrics and geodesics in the space of Kähler metrics on toric varieties

Geodesics on the infinite dimensional symmetric space $\hcal$ of Kähler metrics in a fixed Kähler class on a projective Kähler manifold X are solutions of a homogeneous complex Monge-Ampère 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-Ampère geodesics can be approximated by 1PS geodesics in the symmetric spaces of Bergman metrics. Phong-Sturm proved weak C^0 convergence of Bergman to Monge-Ampère geodesics on a general \kahler manifold. In this article we prove convergence in $C^2(A \times X)$ in the case of toric Kähler metrics, extending our earlier result on $\CP^1$.

math.CV↗

arXiv2008

Number variance of random zeros on complex manifolds

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 $N^{m-1/2} ν_{mm} Vol(\partial U)$, where $ν_{mm}$ 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 $k<m$ random degree-$N$ polynomials on $C^m$. Our results hold more generally for the simultaneous zeros of random holomorphic sections of the $N$-th power of any positive line bundle over any $m$-dimensional compact Kähler manifold.

math.CV↗

arXiv2008

Overcrowding and hole probabilities for random zeros on complex manifolds

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 $δ>0$, the probability that this volume differs by more than $δN$ from its average value is less than $\exp(-C_{δ,U}N^{m+1})$, for some constant $C_{δ,U}>0$. As a consequence, the "hole probability" that a random section does not vanish in U has an upper bound of the form $\exp(-C_{U}N^{m+1})$.

math.CV↗

arXiv2008

On the Spectrum of geometric operators on Kähler manifolds

On a compact Kähler 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 taking these symmetries into account quantum ergodicity holds for the Laplace-Beltrami operator and for the Spin^c-Dirac operators if the unitary frame flow is ergodic. The assumptions for our theorem are known to be satisfied for instance for negatively curved Kähler manifolds of odd complex dimension.

math.DG↗

arXiv2008

Real and complex zeros of Riemannian random waves

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 tube in the complexification of $M$ tends to a limit current.

math.SP↗

arXiv2008

Bergman approximations of harmonic maps into the space of Kahler metrics on toric varieties

We generalize the results of Song-Zelditch on geodesics in spaces of Kahler metrics on toric varieties to harmonic maps of any compact Riemannian manifold with boundary into the space of Kahler metrics on a toric variety. We show that the harmonic map equation can always be solved and that such maps may be approximated in the C^2 topology by harmonic maps into the spaces of Bergman metrics. In particular, WZW maps, or equivalently solutions of a homogeneous Monge-Ampere equation on the product of the manifold with a Riemann surface with S^1 boundary admit such approximations. We also show that the Eells-Sampson flow on the space of Kahler potentials is transformed to the usual heat flow on the space of symplectic potentials under the Legendre transform.

math.DG↗

arXiv2008

The inverse resonance problem for $\Z_2$-symmetric analytic obstacles in the plane

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.

math.SP↗

arXiv2008

Inverse spectral problem for analytic plane domains II: $\Z_2$-symmetric domains

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.

math.SP↗