Weyl Law Research Articles

The Precise Shape of the Eigenvalue Intensity for a Class of Non-Self-Adjoint Operators Under Random Perturbations

We consider a non-self-adjoint h-differential model operator $$P_h$$ in the semiclassical limit ( $$h\rightarrow 0$$ ) subject to small random perturbations. Furthermore, we let the coupling constant $$\delta $$ be $$\mathrm {e}^{-\frac{1}{Ch}}\le \delta \ll h^{\kappa }$$ for constants $$C,\kappa >0$$ suitably large. Let $$\Sigma $$ be the closure of the range of the principal symbol. Previous results on the same model by Hager, Bordeaux-Montrieux and Sjostrand show that if $$\delta \gg \mathrm {e}^{-\frac{1}{Ch}}$$ there is, with a probability close to 1, a Weyl law for the eigenvalues in the interior of the of the pseudospectrum up to a distance $$\gg \left( -h\ln {\delta h}\right) ^{\frac{2}{3}}$$ to the boundary of $$\Sigma $$ . We study the intensity measure of the random point process of eigenvalues and prove an h-asymptotic formula for the average density of eigenvalues. With this we show that there are three distinct regions of different spectral behavior in $$\Sigma $$ : the interior of the pseudospectrum is solely governed by a Weyl law, close to its boundary there is a strong spectral accumulation given by a tunneling effect followed by a region where the density decays rapidly.

Born-Jordan and Weyl Quantizations of the 2D Anisotropic Harmonic Oscillator

We apply the Born-Jordan and Weyl quantization formulas for polynomials in canonical coordinates to the constants of motion of some examples of the superintegrable 2D anisotropic harmonic oscillator. Our aim is to study the behaviour of the algebra of the constants of motion after the different quantization procedures. In the examples considered, we have that the Weyl formula always preserves the original superintegrable structure of the system, while the Born-Jordan formula, when producing different operators than the Weyl's one, does not.

Estimates of the gaps between consecutive eigenvalues of Laplacian

By the calculation of the gap of the consecutive eigenvalues of $\Bbb S^n$ with standard metric, using the Weyl's asymptotic formula, we know the order of the upper bound of this gap is $k^{\frac{1}{n}}.$ We conjecture that this order is also right for general Dirichlet problem of the Laplace operator, which is optimal if this conjecture holds, obviously. In this paper, using new method, we solve this conjecture in the Euclidean space case intrinsically. We think our method is valid for the case of general Riemannian manifolds and give some examples directly.

Effective Confining Potential of Quantum States in Disordered Media.

The amplitude of localized quantum states in random or disordered media may exhibit long-range exponential decay. We present here a theory that unveils the existence of an effective potential which finely governs the confinement of these states. In this picture, the boundaries of the localization subregions for low energy eigenfunctions correspond to the barriers of this effective potential, and the long-range exponential decay characteristic of Anderson localization is explained as the consequence of multiple tunneling in the dense network of barriers created by this effective potential. Finally, we show that Weyl's formula based on this potential turns out to be a remarkable approximation of the density of states for a large variety of one-dimensional systems, periodic or random.

Weyl Type Asymptotics and Bounds for the Eigenvalues of Functional-Difference Operators for Mirror Curves

We investigate Weyl type asymptotics of functional-difference operators associated to mirror curves of special del Pezzo Calabi-Yau threefolds. These operators are \({H(\zeta) = U + U^{-1} + V + \zeta V^{-1}}\) and \({H_{m,n} = U + V + q^{-mn}U^{-m}V^{-n}}\), where \({U}\) and \({V}\) are self-adjoint Weyl operators satisfying \({UV = q^{2}VU}\) with \({q = {\rm e}^{{\rm i}\pi b^{2}}}\), \({b > 0}\) and \({\zeta > 0}\), \({m, n \in \mathbb{N}}\). We prove that \({H(\zeta)}\) and \({H_{m,n}}\) are self-adjoint operators with purely discrete spectrum on \({L^{2}(\mathbb{R})}\). Using the coherent state transform we find the asymptotical behaviour for the Riesz mean \({\sum_{j\ge 1}(\lambda - \lambda_{j})_{+}}\) as \({\lambda \to \infty}\) and prove the Weyl law for the eigenvalue counting function \({N(\lambda)}\) for these operators, which imply that their inverses are of trace class.

Redshift formulas and the Doppler–Fizeau effect

In this paper, we show that redshifts, which appear in some pedagogical examples, can be expressed in terms of the Doppler–Fizeau effect. For this purpose, we use, as suggested by Weyl, the worldline elements of two physical events: the emission and the reception of a monochromatic wave. The redshift in special relativity and its Galilean approximation are derived in a simpler way than is usually done. In general relativity, the cosmological redshift can be obtained with the general Weyl formula in three important cases of gravitational fields, even though the gravitational redshift, due to bodies running away from each other, cannot be reduced to a simple kinematic effect.

An orthogonality relation for a thin family of GL(3) Maass forms

We prove an orthogonality relation for the Fourier–Whittaker coefficients of a thin family of GL(3) Maass forms containing all self-dual forms. This is obtained by analyzing the Kuznetsov trace formula on GL(3) for a certain family of test functions. The method also yields Weyl's law for the same family of Maass forms.

Spectral analysis and the Aharonov-Bohm effect on certain almost-Riemannian manifolds

ABSTRACTWe study spectral properties of the Laplace-Beltrami operator on two relevant almost-Riemannian manifolds, namely the Grushin structures on the cylinder and on the sphere. This operator contains first order diverging terms caused by the divergence of the volume. We get explicit descriptions of the spectrum and the eigenfunctions. In particular in both cases we get a Weyl's law with leading term Elog E. We then study the drastic effect of Aharonov-Bohm magnetic potentials on the spectral properties. Other generalized Riemannian structures including conic and anti-conic type manifolds are also studied. In this case, the Aharonov-Bohm magnetic potential may affect the self-adjointness of the Laplace-Beltrami operator.

Asymptotic spectral gap and Weyl law for Ruelle resonances of open partially expanding maps

We consider a simple model of an open partially expanding map. Its trapped set ${\mathcal{K}}$ in phase space is a fractal set. We first show that there is a well-defined discrete spectrum of Ruelle resonances which describes the asymptotic of correlation functions for large time and which is parametrized by the Fourier component $\unicode[STIX]{x1D708}$ in the neutral direction of the dynamics. We introduce a specific hypothesis on the dynamics that we call ‘minimal captivity’. This hypothesis is stable under perturbations and means that the dynamics is univalued in a neighborhood of ${\mathcal{K}}$. Under this hypothesis we show the existence of an asymptotic spectral gap and a fractal Weyl law for the upper bound of density of Ruelle resonances in the semiclassical limit $\unicode[STIX]{x1D708}\rightarrow \infty$. Some numerical computations with the truncated Gauss map and Bowen–Series maps illustrate these results.

Scaling limit for the kernel of the spectral projector and remainder estimates in the pointwise Weyl law

Let (M, g) be a compact smooth Riemannian manifold. We obtain new off-diagonal estimates as {\lambda} tend to infinity for the remainder in the pointwise Weyl Law for the kernel of the spectral projector of the Laplacian onto functions with frequency at most {\lambda}. A corollary is that, when rescaled around a non self-focal point, the kernel of the spectral projector onto the frequency interval (\lambda, \lambda + 1] has a universal scaling limit as {\lambda} goes to infinity (depending only on the dimension of M). Our results also imply that if M has no conjugate points, then immersions of M into Euclidean space by an orthonormal basis of eigenfunctions with frequencies in (\lambda, \lambda + 1] are embeddings for all {\lambda} sufficiently large.

Smooth Billiards with a Large Weyl Remainder

We prove an estimate from below for the remainder in Weyl's law for smooth star-shaped planar domains admitting an appropriate one-parameter family of periodic billiard trajectories. Examples include ellipses and smooth domains of constant width. Our results confirm a prediction of P. Sarnak who proved a similar statement for surfaces without boundary. The proof is based on the analysis of the singularity in the wave trace, occurring at time T>0T>0 equal to the length of the closed trajectories belonging to the family. Such families of periodic orbits produce large singularities in the wave trace, which in turn imply estimates from below for the absolute value of the Weyl remainder averaged over a spectral window. We also obtain lower bounds for the error term in higher dimensions. In this case, the main contribution to the Weyl remainder typically comes from the singularity at zero of the wave trace. However, for certain domains, such as the Euclidean ball, the dimension of the family of periodic trajectories is large enough to dominate the contribution of the singularity at zero.

A general simple relative trace formula

In this paper, we prove a general simple relative trace formula. As an application, we prove a relative analogue of the Weyl law.

Scalar curvature for noncommutative four-tori

In this paper we study the curved geometry of noncommutative 4-tori \mathbb{T}_{\theta}^4 . We use a Weyl conformal factor to perturb the standard volume form and obtain the Laplacian that encodes the local geometric information. We use Connes' pseudodifferential calculus to explicitly compute the terms in the small time heat kernel expansion of the perturbed Laplacian which correspond to the volume and scalar curvature of \mathbb{T}_{\theta}^4 . We establish the analogue of Weyl's law, define a noncommutative residue, prove the analogue of Connes' trace theorem, and find explicit formulas for the local functions that describe the scalar curvature of \mathbb{T}_{\theta}^4 . We also study the analogue of the Einstein-Hilbert action for these spaces and show that metrics with constant scalar curvature are critical for this action.

The six-vertex model and deformations of the Weyl character formula

We use statistical mechanics--variants of the six-vertex model in the plane studied by means of the Yang---Baxter equation--to give new deformations of Weyl's character formula for classical groups of Cartan type B, C, and D, and a character formula of Proctor for type BC. In each case, the corresponding Boltzmann weights are associated with the free fermion point of the six-vertex model. These deformations add to the earlier known examples in types A and C by Tokuyama and Hamel-King, respectively. A special case for classical types recovers deformations of the Weyl denominator formula due to Okada.

Asymptotic Behavior of the Spectrum of Pseudodifferential Operators of Variable Order

We consider compact selfadjoint pseudodifferential operators under the assumption that the decay order of symbols with respect to ξ depends on a point x. We show that the asymptotic Weyl formula is valid for such operators. Bibliography: 12 titles.

Equivariant local scaling asymptotics for smoothed Töplitz spectral projectors

Let X be the unit circle bundle of a positive line bundle on a Hodge manifold. We study the local scaling asymptotics of the smoothed spectral projectors associated with a first order elliptic Töplitz operator T on X , possibly in the presence of Hamiltonian symmetries. The resulting expansion is then used to give a local derivation of an equivariant Weyl law. It is not required that T be invariant under the structure circle action, that is, T needn't be a Berezin–Töplitz operator.

Sharp Weyl estimates for tensor products of pseudodifferential operators

We study the asymptotic behavior of the counting function of tensor products of operators, in the cases where the factors are either pseudodifferential operators on closed manifolds or pseudodifferential operators of Shubin type on \({\mathbb {R}}^{n}\), respectively. We obtain, in particular, the sharpness of the remainder term in the corresponding Weyl formulae, which we prove by means of the analysis of some explicit examples.

Asymptotics of Linear Waves and Resonances with Applications to Black Holes

We apply the results of arXiv:1301.5633 to describe asymptotic behavior of linear waves on stationary Lorentzian metrics with r-normally hyperbolic trapped sets, in particular Kerr and Kerr-de Sitter metrics with |a|<M and M\Lambda a << 1. We prove that if the initial data is localized at frequencies \lambda >> 1, then the energy norm of the solution is bounded by O(\lambda^{1/2} exp(-(\nu_min - \epsilon)t/2) + \lambda^(-\infty)), for t < C log\lambda, where \nu_min is a natural dynamical quantity. The key tool is a microlocal projector splitting the solution into a component with controlled rate of exponential decay and an O(\lambda exp(-(\nu_min -\epsilon)t) + \lambda^(-\infty)) remainder; this splitting can be viewed as an analog of resonance expansion. Moreover, for the Kerr-de Sitter case we study quasi-normal modes; under a dynamical pinching condition, a Weyl law in a band holds.

Quantum signatures of classical multifractal measures.

A clear signature of classical chaoticity in the quantum regime is the fractal Weyl law, which connects the density of eigenstates to the dimension D(0) of the classical invariant set of open systems. Quantum systems of interest are often partially open (e.g., cavities in which trajectories are partially reflected or absorbed). In the corresponding classical systems D(0) is trivial (equal to the phase-space dimension), and the fractality is manifested in the (multifractal) spectrum of Rényi dimensions D(q). In this paper we investigate the effect of such multifractality on the Weyl law. Our numerical simulations in area-preserving maps show for a wide range of configurations and system sizes M that (i) the Weyl law is governed by a dimension different from D(0)=2, and (ii) the observed dimension oscillates as a function of M and other relevant parameters. We propose a classical model that considers an undersampled measure of the chaotic invariant set, explains our two observations, and predicts that the Weyl law is governed by a nontrivial dimension D(asymptotic)<D(0) in the semiclassical limit M→∞.

Eigenvalues of the Laplacian on a compact manifold with density

In this paper, we study the spectrum of the weighted Laplacian (also called Bakry-Emery or Witten Laplacian) $L_\sigma$ on a compact, connected, smooth Riemannian manifold $(M,g)$ endowed with a measure $\sigma dv_g$. First, we obtain upper bounds for the $k-$th eigenvalue of $L_{\sigma}$ which are consistent with the power of $k$ in Weyl's formula. These bounds depend on integral norms of the density $\sigma$, and in the second part of the article, we give examples showing that this dependence is, in some sense, sharp. As a corollary, we get bounds for the eigenvalues of Laplace type operators, such as the Schrodinger operator or the Hodge Laplacian on $p-$forms. In the special case of the weighted Laplacian on the sphere, we get a sharp inequality for the first nonzero eigenvalue which extends Hersch's inequality.