Dmitry Jakobson

Spectra of elements in the group ring of SU(2)

We present a new method for establishing the “gap” property for finitely generated subgroups of SU(2), providing an elementary solution of Ruziewicz problem on S2 as well as giving many new examples of finitely generated subgroups of SU(2) with an explicit gap. The distribution of the eigenvalues of the elements of the group ring R[SU(2)] in the N-th irreducible representation of SU(2) is also studied. Numerical experiments indicate that for a generic (in measure) element of R[SU(2)], the “unfolded” consecutive spacings distribution approaches the GOE spacing law of random matrix theory (for N even) and the GSE spacing law (for N odd) as N→∞; we establish several results in this direction. For certain special “arithmetic” (or Ramanujan) elements of R[SU(2)] the experiments indicate that the unfolded consecutive spacing distribution follows Poisson statistics; we provide a sharp estimate in that direction.

Eigenvalue Spacings for Regular Graphs

We carry out a numerical study of fluctuations in the spectra of regular graphs. Our experiments indicate that the level spacing distribution of a generic k-regular graph approaches that of the Gaussian Orthogonal Ensemble of random matrix theory as we increase the number of vertices. A review of the basic facts on graphs and their spectra is included.

Extremal Metric for the First Eigenvalue on a Klein Bottle

The first eigenvalue of the Laplacian on a surface can be viewed as a functional on the space of Riemannian metrics of a given area. Critical points of this functional are called extremal metrics. The only known extremal metrics are a round sphere, a standard projective plane, a Clifford torus and an equilateral torus. We construct an extremal metric on a Klein bottle. It is a metric of revolution, admitting a minimal isometric embedding into a sphere S 4 by the first eigenfunctions. Also, this Klein bottle is a bipolar surface for Lawsons �3,1-torus. We conjecture that an extremal metric for the first eigenvalue on a Klein bottle is unique, and hence it provides a sharp upper bound for �1 on a Klein bottleofagivenarea. Wepresentnumericalevidenceandprovethefirstresultstowardsthisconjecture.

How large can the first eigenvalue be on a surface of genus two

AbstractSharp upper bounds for the first eigenvalue of the Laplacian on asurface of a fixed area are known only in genera zero and one. Weinvestigate the genus two case and conjecture that the first eigenvalue ismaximized on a singular surface which is realized as a double branchedcovering over a sphere. The six ramification points are chosen in sucha way that this surface has a complex structure of the Bolza surface.We prove that our conjecture follows from a lower bound on the firsteigenvalue of a certain mixed Dirichlet-Neumann boundary value problemon a half-disk. The latter can be studied numerically, and we presentconclusive evidence supporting the conjecture. Keywords: Laplacian, first eigenvalue, surface of genus two, mixed boundaryvalue problem. ∗ Department of Mathematics and Statistics, McGill University, 805 Sherbrooke Str. West,Montreal, QC H3A 2K6, Canada; e-mail jakobson@math.mcgill.ca † Department of Mathematics, Heriot-Watt University, United Kingdom; e-mailM.Levitin@ma.hw.ac.uk

Classical Limits of Eigenfunctions for Some Completely Integrable Systems

We give an overview of some old results on weak* limits of eigenfunctions and prove some new ones. We first show that on M = (S n , can) every probability measure on S * M which is invariant under the geodesic flow and time reversal is a weak* limit of a sequence of Wigner measures corresponding to eigenfunctions of Δ. We next show that joint eigenfunctions of Δ and a single Hecke operator on S n cannot scar on a single closed geodesic. We finally use the estimates of [Z3] on the rate of quantum ergodicity to prove that adding a ψDO of order - n + 2 doesn’t change the level spacings distribution of Δ (if the former is well defined) on a compact negatively curved manifold of dimension n. In dimension two this shows that the level spacings distributions of quantizations of certain Hamiltonians do not depend on the quantization.

Geometric and Spectral Analysis

Smooth Kähler–Einstein metrics have been studied for the past 80 years. More recently, singular Kähler–Einstein metrics have emerged as objects of intrinsic interest, both in differential and algebraic geometry, as well as a powerful tool in better understanding their smooth counterparts. This article is mostly a survey of some of these developments.

High Energy Limits of Laplace-Type and Dirac-Type EigenFunctions and Frame Flows

We relate high-energy limits of Laplace-type and Dirac-type operators to frame flows on the corresponding manifolds, and show that the ergodicity of frame flows implies quantum ergodicity in an appropriate sense for those operators. Observables for the corresponding quantum systems are matrix-valued pseudodifferential operators and therefore the system remains non-commutative in the high-energy limit. We discuss to what extent the space of stationary high-energy states behaves classically.

A Law of Large Numbers for the Zeroes of Heine–Stieltjes Polynomials

We determine the limiting density of the zeroes of Heine–Stieltjes polynomials (or of any set of points satisfying the conclusion of Heine–Stieltjes Theorem) in the thermodynamic limit and use this to prove a strong law of large numbers for the zeroes.

Tubular neighborhoods of nodal sets and diophantine approximation

We give upper and lower bounds on the volume of a tubular neighborhood of the nodal set of an eigenfunction of the Laplacian on a real analytic closed Riemannian manifold

Lower bounds for the spectral function and for the remainder in local Weyl’s law on manifolds

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