Mathematics

This paper is devoted to the asymptotic behavior of all eigenvalues of Symmetric (in general non Hermitian) Toeplitz matrices with moderately smooth symbols which trace out a simple loop on the complex plane line as the dimension of the matrices increases to infinity. The main result describes the asymptotic structure of all eigenvalues. The constructed expansion is uniform with respect to the number of eigenvalues. Keywords: Toeplitz matrices, eigenvalues, asymptotic expansions

We consider the family h ^ μ := Δ ^ Δ ^ −μ v ^ ,μ∈R, of discrete Schrödinger-type operators in one-dimensional lattice Z , where Δ ^ is the discrete Laplacian and v ^ is of zero-range. We prove that for any μ≠0 the discrete spectrum of h ^ μ is a singleton {e(μ)}, and e(μ)<0 for μ>0 and e(μ)>4 for μ<0. Moreover, we study the properties of e(μ) as a function of μ, in particular, we find the asymptotics of e(μ) as μ↘0 and μ↗0.

We consider the three-dimensional sloshing problem on a triangular prism whose angles with the sloshing surface are of the form π 2q , where q is an integer. We are interested in finding a two-term asymptotic expansion of the eigenvalue counting function. When both angles are π 4 , we compute the exact value of the second term. As for the general case, we conjecture an asymptotic expansion by constructing quasimodes for the problem and computing the counting function of the related quasi-eigenvalues. These quasimodes come from solutions of the sloping beach problem and correspond to two kinds of waves, edge waves and surface waves. We show that the quasi-eigenvalues are exponentially close to real eigenvalues of the sloshing problem. The asymptotic expansion of their counting function is closely related to a lattice counting problem inside a perturbed ellipse where the perturbation is in a sense random. The contribution of the angles can then be detected through that perturbation.

We estimate the magnetic Laplacian energy norm in appropriate planar domains under a weak regularity hypothesis on the magnetic field. Our main contribution is an averaging estimate, valid in small cells, allowing us to pass from non-uniform to uniform magnetic fields. As a matter of application, we derive new upper and lower bounds of the lowest eigenvalue of the Dirichlet Laplacian which match in the regime of large magnetic field intensity. Furthermore, our averaging technique allows us to estimate the non-linear Ginzburg-Landau energy, and as a byproduct, yields a non-Gaussian trial state for the Dirichlet magnetic Laplacian.

In this expository work, we collect some background results and give a short proof of the following theorem: periodic Jacobi matrices on Z d exhibit strong ballistic motion.

In this paper we show that, with probability 1, a random Beltrami field exhibits chaotic regions that coexist with invariant tori of complicated topologies. The motivation to consider this question, which arises in the study of stationary Euler flows in dimension 3, is V.I. Arnold's 1965 conjecture that a typical Beltrami field exhibits the same complexity as the restriction to an energy hypersurface of a generic Hamiltonian system with two degrees of freedom. The proof hinges on the obtention of asymptotic bounds for the number of horseshoes, zeros, and knotted invariant tori and periodic trajectories that a Gaussian random Beltrami field exhibits, which we obtain through a nontrivial extension of the Nazarov--Sodin theory for Gaussian random monochromatic waves and the application of different tools from the theory of dynamical systems, including KAM theory, Melnikov analysis and hyperbolicity. Our results hold both in the case of Beltrami fields on R 3 and of high-frequency Beltrami fields on the 3-torus.

Let G be a Lie group with Lie algebra g and let π be a unitary representation of G realized on a reproducing kernel Hilbert space. We use Berezin quantization to study spectral measures associated with operators −idπ(X) for X∈g . As an application, we show how results about contractions of Lie group representations give rise to results on convergence of sequences of spectral measures. We give some examples including contractions of SU(1,1) and SU(2) to the Heisenberg group.

The principal aim of this paper is to employ Bessel-type operators in proving the inequality ??? 0 dx| f ??(x) | 2 ??1 4 ??? 0 dx |f(x) | 2 sin 2 (x) + 1 4 ??? 0 dx|f(x) | 2 ,f??H 1 0 ((0,?)), where both constants 1/4 appearing in the above inequality are optimal. In addition, this inequality is strict in the sense that equality holds if and only if f?? . This inequality is derived with the help of the exactly solvable, strongly singular, Dirichlet-type Schrödinger operator associated with the differential expression ? s =??d 2 d x 2 + s 2 ??1/4) sin 2 (x) ,s?�[0,??,x??0,?). The new inequality represents a refinement of Hardy's classical inequality ??? 0 dx| f ??(x) | 2 ??1 4 ??? 0 dx |f(x) | 2 x 2 ,f??H 1 0 ((0,?)), it also improves upon one of its well-known extensions in the form ??? 0 dx| f ??(x) | 2 ??1 4 ??? 0 dx |f(x) | 2 d (0,?) (x ) 2 ,f??H 1 0 ((0,?)), where d (0,?) (x) represents the distance from x??0,?) to the boundary {0,?} of (0,?) .

Under certain assumptions (including d\ge 2) we prove that the spectrum of a scalar operator in \mathscr{L}^2(\mathbb{R}^d) \begin{equation*} A_\varepsilon (x,hD)= A^0(hD) + \varepsilon B(x,hD), \end{equation*} covers interval (\tau-\epsilon,\tau+\epsilon) , where A^0 is an elliptic operator and B(x,hD) is a periodic perturbation, \varepsilon=O(h^\varkappa) , \varkappa>0 . Further, we consider generalizations.

We estimate the bottom of the L 2 spectrum of the Laplacian on locally symmetric spaces in terms of the critical exponents of appropriate Poincaré series. Our main result is the higher rank analog of a characterization due to Elstrodt, Patterson, Sullivan and Corlette in rank one. It improves upon previous results obtained by Leuzinger and Weber in higher rank.