Motivated by recent interest in the spectrum of the Laplacian of incomplete surfaces with isolated conical singularities, we consider more general incomplete m-dimensional manifolds with singularities on sets of codimension at least 2. With certain restrictions on the metric, we establish that the spectrum is discrete and satisfies Weyl’s asymptotic formula. 1. Discreteness of the Spectrum When...

For the random eigenvalues with density corresponding to the Jacobi ensemble c · ∏

This paper addresses the issue of quantifying the frequency domain accuracy of ARMA spectral estimates as dictated by the Cramér–Rao Lower Bound (CRLB). Classical work in this area has led to expressions that are asymptotically exact as both data length and model order tend to infinity, although they are commonly used in finite model order and finite data length settings as approximations. More...

Consider a n × n matrix from the Gaussian Unitary Ensemble (GUE). Given a finite collection of bounded disjoint real Borel sets (∆i,n, 1 ≤ i ≤ p), properly rescaled, and eventually included in any neighbourhood of the support of Wigner’s semi-circle law, we prove that the related counting measures (Nn(∆i,n), 1 ≤ i ≤ p), where Nn(∆) represents the number of eigenvalues within ∆, are asymptotical...

Classical spectral analysis in statistics considers decomposition of stationary time series into sinusoidal components. The autocovariance and the spectrum are fundamental elements for analyzing a given time series both in time and frequency domain. However, in practice one frequently observes nonstationary time series. In order to apply spectral analysis to these processes, an extension of the...

We prove the WKB asymptotic behavior of solutions of the differential equation −d2u/dx2 + V (x)u = Eu for a.e. E > A where V = V1 + V2, V1 ∈ L(R), and V2 is bounded from above with A = lim supx→∞ V (x), while V ′ 2(x) ∈ L(R), 1 ≤ p < 2. These results imply that Schrödinger operators with such potentials have absolutely continuous spectrum on (A,∞). We also establish WKB asymptotic behavior of s...

In this note, we are concerned with the asymptotic approximation of a class of double integrals which can be represented as an angular spectrum superposition. These double integrals typically appear in electromagnetic scattering problems. Based on the synthetic manipulation of the method of steepest descent path, approximate expressions of the double integrals are derived in terms of the leadin...

We introduce a new type of local and microlocal asymptotic analysis in algebras of generalized functions, based on the presheaf properties of those algebras and on the properties of their elements with respect to a regularizing parameter. Contrary to the more classical frequential analysis based on the Fourier transform, we can describe a singular asymptotic spectrum which has good properties w...

It is shown, by means of a simple specific example, that for integrable systems it is possible to build up approximate eigenfunctions, called asymptotic eigenfunctions, which are concentrated as much as one wants to a classical trajectory and have a lifetime as long as one wants. These states are directly related to the presence of shell structures in the quantal spectrum of the system. It is a...

Let Γ be a finitely generated group with a given word metric. The asymptotic density of elements in Γ that have a particular property P is the limit, as r → ∞, of the proportion of elements in the ball of radius r which have the property P . We obtain a formula to compute the asymptotic density of finite-order elements in any virtually nilpotent group. Further, we show that the spectrum of numb...