Weyl Law Research Articles

Hybrid Finite Element--Spectral Method for the Fractional Laplacian: Approximation Theory and Efficient Solver

A numerical scheme is presented for approximating fractional order Poisson problems in two and three dimensions. The scheme is based on reformulating the original problem posed over $\Omega$ on the extruded domain $\mathcal{C}=\Omega\times[0,\infty)$ following Caffarelli and Silvestre (2007). The resulting degenerate elliptic integer order PDE is then approximated using a hybrid FEM-spectral scheme. Finite elements are used in the direction parallel to the problem domain $\Omega$, and an appropriate spectral method is used in the extruded direction. The spectral part of the scheme requires that we approximate the true eigenvalues of the integer order Laplacian over $\Omega$. We derive an a priori error estimate which takes account of the error arising from using an approximation in place of the true eigenvalues. We further present a strategy for choosing approximations of the eigenvalues based on Weyl's law and finite element discretizations of the eigenvalue problem. The system of linear algebraic equations arising from the hybrid FEM-spectral scheme is decomposed into blocks which can be solved effectively using standard iterative solvers such as multigrid and conjugate gradient. Numerical examples in two and three dimensions show that the approach is quasi-optimal in terms of complexity.

On sums of eigenvalues of elliptic operators on manifolds

We use the averaged variational principle introduced in a recent article on graph spectra [10] to obtain upper bounds for sums of eigenvalues of several partial differential operators of interest in geometric analysis, which are analogues of Kröger’s bound for Neumann spectra of Laplacians on Euclidean domains [15]. Among the operators we consider are the Laplace–Beltrami operator on compact subdomains of manifolds. These estimates become more explicit and asymptotically sharp when the manifold is conformal to homogeneous spaces (here extending a result of Strichartz [26] with a simplied proof). In addition we obtain results for the Witten Laplacian on the same sorts of domains and for Schrödinger operators with confining potentials on infinite Euclidean domains. Our bounds have the sharp asymptotic form expected from the Weyl law or classical phase-space analysis. Similarly sharp bounds for the trace of the heat kernel follow as corollaries.

An estimate for the average number of common zeros of Laplacian eigenfunctions

On a compact Riemannian manifold $M$ of dimension $n$, we consider $n$ eigenfunctions of the Laplace operator $\Delta $ with eigenvalue $\lambda$. If $M$ is homogeneous under a compact Lie group preserving the metric then we prove that the average number of common zeros of $n$ eigenfunctions does not exceed $c(n)\lambda^{n/2}{\rm vol}\,M$, the expression known from the celebrated Weyl's law. Moreover, if the isotropy representation is irreducible, then the estimate turns into equality. The constant $c(n)$ is explicitly given. The method of proof is based on the application of Crofton's formula for the sphere.

Fractal Weyl laws and wave decay for general trapping

We prove a Weyl upper bound on the number of scattering resonances in strips for manifolds with Euclidean infinite ends. In contrast with previous results, we do not make any strong structural assumptions on the geodesic flow on the trapped set (such as hyperbolicity) and instead use propagation statements up to the Ehrenfest time. By a similar method we prove a decay statement with high probability for linear waves with random initial data. The latter statement is related heuristically to the Weyl upper bound. For geodesic flows with positive escape rate, we obtain a power improvement over the trivial Weyl bound and exponential decay up to twice the Ehrenfest time.

The equivariant spectral function of an invariant elliptic operator. Lp-bounds, caustics, and concentration of eigenfunctions

Let M be a compact boundaryless Riemannian manifold, carrying an effective and isometric action of a compact Lie group G, and P0 an invariant elliptic classical pseudodifferential operator on M. Using Fourier integral operator techniques, we prove a local Weyl law with remainder estimate for the equivariant (or reduced) spectral function of P0 for each isotypic component in the Peter–Weyl decomposition of L2(M), generalizing work of Avacumovič, Levitan, and Hörmander. From this we deduce a generalized Kuznecov sum formula for periods of G-orbits, and recover the local Weyl law for orbifolds shown by Stanhope and Uribe. Relying on recent results on singular equivariant asymptotics of oscillatory integrals, we further characterize the caustic behavior of the reduced spectral function near singular orbits, which allows us to give corresponding point-wise bounds for clusters of eigenfunctions in specific isotypic components. In case that G acts on M without singular orbits, we are able to deduce hybrid Lp-bounds for 2≤p≤∞ in the eigenvalue and isotypic aspect that improve on the classical estimates of Seeger and Sogge for generic eigenfunctions. Our results are sharp in the eigenvalue aspect, but not in the isotypic aspect, and reduce to the classical ones in the case G={e}.

Weyl formula for the negative dissipative eigenvalues of Maxwell’s equations

Let $$V(t) = e^{tG_b},\, t \ge 0,$$ be the semigroup generated by Maxwell’s equations in an exterior domain $$\Omega \subset {\mathbb R}^3$$ with dissipative boundary condition $$E_{tan}- \gamma (x) (\nu \wedge B_{tan}) = 0, \gamma (x) > 0, \forall x \in \Gamma = \partial \Omega .$$ We study the case when $$\Omega = \{x \in {\mathbb R}^3:\, |x| > 1\}$$ and $$\gamma \ne 1$$ is a constant. We establish a Weyl formula for the counting function of the negative real eigenvalues of $$G_b.$$

Asymptotics of Spectra of Compact Pseudodifferential Operators with Nonsmooth Symbols with Respect to Spatial Variables

We consider compact selfadjoint pseudodifferential operators with symbols that are not smooth with respect to x on a fixed set. We obtain conditions for the validity of the Weyl formula for the spectral asymptotics and apply the results to some class of pseudodifferential operators.

Koszul complexes, Birkhoff normal form and the magnetic Dirac operator

We consider the semi-classical Dirac operator coupled to a magnetic potential on a large class of manifolds including all metric contact manifolds. We prove a sharp local Weyl law and a bound on its eta invariant. In the absence of a Fourier integral parametrix, the method relies on the use of almost analytic continuations combined with the Birkhoff normal form and local index theory.

Counting statistics of chaotic resonances at optical frequencies: Theory and experiments.

A deformed dielectric microcavity is used as an experimental platform for the analysis of the statistics of chaotic resonances, in the perspective of testing fractal Weyl laws at optical frequencies. In order to surmount the difficulties that arise from reading strongly overlapping spectra, we exploit the mixed nature of the phase space at hand, and only count the high-Q whispering-gallery modes (WGMs) directly. That enables us to draw statistical information on the more lossy chaotic resonances, coupled to the high-Q regular modes via dynamical tunneling. Three different models [classical, Random-Matrix-Theory (RMT) based, semiclassical] to interpret the experimental data are discussed. On the basis of least-squares analysis, theoretical estimates of Ehrenfest time, and independent measurements, we find that a semiclassically modified RMT-based expression best describes the experiment in all its realizations, particularly when the resonator is coupled to visible light, while RMT alone still works quite well in the infrared. In this work we reexamine and substantially extend the results of a short paper published earlier [L. Wang et al., Phys. Rev. E 93, 040201(R) (2016)2470-004510.1103/PhysRevE.93.040201].

Sup norms of Cauchy data of eigenfunctions on manifolds with concave boundary

ABSTRACTWe prove that the Cauchy data of Dirichlet or Neumann Δ- eigenfunctions of Riemannian manifolds with concave (diffractive) boundary can only achieve maximal sup norm bounds if there exists a self-focal point on the boundary, i.e., a point at which a positive measure of geodesics leaving the point return to the point. In the case of real analytic Riemannian manifolds with real analytic boundary, maximal sup norm bounds on boundary traces of eigenfunctions can only be achieved if there exists a point on the boundary at which all geodesics loop back. As an application, the Dirichlet or Neumann eigenfunctions of Riemannian manifolds with concave boundary and non-positive curvature never have eigenfunctions whose boundary traces achieve maximal sup norm bounds. The key new ingredient is the Melrose–Taylor diffractive parametrix and Melrose’s analysis of the Weyl law.

$$C^\infty $$ C ∞ Scaling Asymptotics for the Spectral Projector of the Laplacian

This article concerns new off-diagonal estimates on the remainder and its derivatives in the pointwise Weyl law on a compact n-dimensional Riemannian manifold. As an application, we prove that near any non-self-focal point, the scaling limit of the spectral projector of the Laplacian onto frequency windows of constant size is a normalized Bessel function depending only on n.

Weyl formulae for the Robin Laplacian in the semiclassical limit

This paper is devoted to establish semiclassical Weyl formulae for the Robin Laplacian on smooth domains in any dimension. Theirs proofs are reminiscent of the Born-Oppenheimer method.

Quantum ergodicity and symmetry reduction

We study the spectral and ergodic properties of Schrödinger operators on a compact connected Riemannian manifold M without boundary in case that the examined system possesses certain symmetries. More precisely, if M carries an isometric and effective action of a compact connected Lie group G, we prove a generalized equivariant version of the semiclassical Weyl law with an estimate for the remainder, using a semiclassical functional calculus for h-dependent functions and relying on recent results on singular equivariant asymptotics. We then deduce an equivariant quantum ergodicity theorem under the assumption that the symmetry-reduced Hamiltonian flow on the principal stratum of the singular symplectic reduction of M is ergodic. In particular, we obtain an equivariant version of the Shnirelman–Zelditch–Colin-de-Verdière theorem, as well as a representation theoretic equidistribution theorem. If M/G is an orbifold, similar results were recently obtained by Kordyukov. When G is trivial, one recovers the classical results.

On the Weyl Law for Quantum Graphs

Roughly speaking, the Weyl law describes the asymptotic distribution of eigenvalues of the Laplacian that can be attached to different objects and can be analyzed in different settings. The form of the remainder term in the Weyl law is very significant in applications, and a power-saving exponent in the remainder term is appreciated. We are dealing with Laplacian defined on compact metric graph with general self-adjoint boundary conditions. The main purpose of this paper is to present application of the special form of the Tauberian theorem for the Laplace transform to the suitably transformed trace formula in the above-mentioned quantum graphs setting. The key feature of our method is that it produces a power-saving form of the reminder term and hence represents improvement in classical methods, which may be applied in other settings as well. The obtained form of the Weyl law is with the power saving of 1 / 3 in the remainder term.

The $\Theta$ Function and the Weyl Law on Manifolds Without Conjugate Points

We prove that the usual \Theta function on a Riemannian manifold without conjugate points is uniformly bounded from below. This extends a result of Green in two dimensions. We deduce that the Bérard remainder in the Weyl law is valid for a manifold without conjugate points, without any restriction on the dimension.

NONCLASSICAL SPECTRAL ASYMPTOTICS AND DIXMIER TRACES: FROM CIRCLES TO CONTACT MANIFOLDS

We consider the spectral behavior and noncommutative geometry of commutators$[P,f]$, where$P$is an operator of order 0 with geometric origin and$f$a multiplication operator by a function. When$f$is Hölder continuous, the spectral asymptotics is governed by singularities. We study precise spectral asymptotics through the computation of Dixmier traces; such computations have only been considered in less singular settings. Even though a Weyl law fails for these operators, and no pseudodifferential calculus is available, variations of Connes’ residue trace theorem and related integral formulas continue to hold. On the circle, a large class of nonmeasurable Hankel operators is obtained from Hölder continuous functions$f$, displaying a wide range of nonclassical spectral asymptotics beyond the Weyl law. The results extend from Riemannian manifolds to contact manifolds and noncommutative tori.

The space of cycles, a Weyl law for minimal hypersurfaces and Morse index estimates

The space of cycles, a Weyl law for minimal hypersurfaces and Morse index estimates

SPECTRAL ZETA FUNCTIONS FOR THE QUANTUM RABI MODELS

We introduce the Hurwitz-type spectral zeta functions for the quantum Rabi models, and give their meromorphic continuation to the whole complex plane with only one simple pole at$s=1$. As an application, we give the Weyl law for the quantum Rabi models. As a byproduct, we also give a rationality of Rabi–Bernoulli polynomials introduced in this paper.

Tunneling for the Robin Laplacian in smooth planar domains

We study the low-lying eigenvalues of the semiclassical Robin Laplacian in a smooth planar domain with bounded boundary which is symmetric with respect to an axis. In the case when the curvature of the boundary of the domain attains its maximum at exactly two points away from the axis of symmetry, we establish an explicit asymptotic formula for the splitting of the first two eigenvalues. This is a rigorous derivation of the semiclassical tunneling effect induced by the domain’s geometry. Our approach is close to the Born–Oppenheimer one and yields, as a byproduct, a Weyl formula of independent interest.

The semiclassical zeta function for geodesic flows on negatively curved manifolds

We consider the semi-classical (or Gutzwiller-Voros) zeta function for $C^\infty$ contact Anosov flows. Analyzing the spectrum of transfer operators associated to the flow, we prove, for any $\tau>0$, that its zeros are contained in the union of the $\tau$-neighborhood of the imaginary axis, $|\Re(s)|<\tau$, and the region $\Re(s)<-\chi_0+\tau$, up to finitely many exceptions, where $\chi_0>0$ is the hyperbolicity exponent of the flow. Further we show that the zeros in the neighborhood of the imaginary axis satisfy an analogue of the Weyl law.