arXiv research

Researchers prove Weyl laws for Schrödinger operators on noncompact manifolds.

In this paper, we will prove the Weyl's law for the asymptotic formula of Dirichlet eigenvalues on metric measure spaces with generalized Ricci curvature bounded from below.

Study spectral properties of sub-Riemannian Laplacians, proving quantum ergodicity and heat kernel asymptotics.

New criterion for Weyl law on Riemannian manifolds without standard assumptions.

Combines noncommutative geometry and spectral theory for new Weyl laws.

We prove an off-diagonal expansion for a Toeplitz operator with an indicator function.

The study improves bounds on the number of closed geodesics and logarithmic improvements in the Weyl law.

We prove the analogue of Weyl's law for a noncommutative Riemannian manifold, namely the noncommutative two torus $\mathbb{T}_θ^2$ equipped with a general translation invariant conformal structure and a Weyl conformal factor. This is achieved by studying the asymptotic distribution of the eigenvalues of the perturbed L…

Let $M$ be a closed Riemannian manifold carrying an effective and isometric action of a compact connected Lie group $G$. We derive a refined remainder estimate in the stationary phase approximation of certain oscillatory integrals on $T^\ast M \times G$ with singular critical sets that were examined previously in order…

We obtain an estimate from below for the remainder in Weyl's law on negatively curved surfaces. In the constant curvature case, such a bound was proved independently by Hejhal and Randol in 1976 using the Selberg zeta function techniques. Our approach works in arbitrary negative curvature, and is based on wave trace as…

We obtain asymptotic lower bounds for the spectral function of the Laplacian and for the remainder in local Weyl's law on manifolds. In the negatively curved case, thermodynamic formalism is applied to improve the estimates. Key ingredients of the proof include the wave equation parametrix, a pretrace formula and the D…

The Planck mass can be derived from gravitational potential behavior in compactifications.

Generic density of equivariant min-max hypersurfaces in Riemannian manifolds.

The paper extends Weyl's law to CROSSes, showing sharpness and polynomial improvement.

We study the asymptotic growth of the eigenvalues of the Laplace-Beltrami operator on singular Riemannian manifolds, where all geometrical invariants appearing in classical spectral asymptotics are unbounded, and the total volume can be infinite. Under suitable assumptions on the curvature blow-up, we show how the sing…

Researchers found a Weyl law for Liouville quantum gravity eigenvalues.

Derives Weyl law for volume spectrum using parametric inequalities.

Proves the Weyl law for 1-cycles in manifolds.

We study the heat trace for both the drifting Laplacian as well as Schrödinger operators on compact Riemannian manifolds. In the case of a finite regularity potential or weight function, we prove the existence of a partial (six term) asymptotic expansion of the heat trace for small times as well as a suitable remainder…

The study shows how geometric Weyl bulk-density exponent rigidifies spectral encodings in O-regularly varying classes.

Given $M$ a Riemannian manifold with (possibly empty) boundary, we show that its volume spectrum $\{ω_p(M)\}_{p\in\mathbb{N}}$ satisfies a Weyl law that was conjectured by Gromov.

We show that a Weyl law holds for the variational spectrum of the $p$-Laplacian. More precisely, let $(λ_i)_{i=1}^\infty$ be the variational spectrum of $Δ_p$ on a closed Riemannian manifold $(X,g)$ and let $N(λ) = \#\{i:\, λ_i < λ\}$ be the associated counting function. Then we have a Weyl law $N(λ) \sim c \operatorna…

The short-time heat kernel expansion of elliptic operators provides a link between local and global features of classical geometries. For many geometric structures related to (non-)involutive distributions, the natural differential operators tend to be Rockland, hence hypoelliptic. In this paper we establish a universa…

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 …

New Weyl's laws discovered for compact spaces with Ricci curvature bounds.

The study provides a formula for the volume of leaf spaces of certain foliations.

Let $X$ be a locally symmetric space defined by a simple Chevalley group $G$ and a congruence subgroup of $G(\mathbb Q)$. In this generality, the Weyl law for $X$ was proved by Lindenstrauss--Venkatesh. In the case where $G$ is simply connected, we sharpen their result by giving a power saving estimate for the remainde…

In this paper, we prove pointwise convergence of heat kernels for mGH-convergent sequences of $RCD^*(K,N)$-spaces. We obtain as a corollary results on the short-time behavior of the heat kernel in $RCD^*(K,N)$-spaces. We use then these results to initiate the study of Weyl's law in the $RCD$ setting

The paper shows equidistribution of geodesics and nets on manifolds.

Under suitable invertibility hypothesis, the spectrum of the Dirac operator on certain open spin Riemannian manifolds is discrete, and obeys a growth law depending qualitatively on the (in)finiteness of the volume.

Study spectral analysis on lens spaces, proving isospectral lens spaces with prime order fundamental groups.

Defines half-volume spectrum for manifolds and proves Weyl law holds.

Harish-Chandra's volume formula shows that the volume of a flag manifold $G/T$, where the measure is induced by an invariant inner product on the Lie algebra of $G$, is determined up to a scalar by the algebraic properties of $G$. This article explains how to deduce Harish-Chandra's formula from Weyl's law by utilizing…

Upper bound found for Steklov eigenvalues counting function.

We study an asymptotic Dirichlet problem for Weyl structures on asymptotically hyperbolic manifolds. By the bulk-boundary correspondence, or more precisely by the Fefferman-Graham theorem on Poincaré metrics, this leads to a natural extension of the notion of Branson's $Q$-curvature to Weyl structures on even-dimension…

Our topological setting is a smooth compact manifold of dimension two or higher with smooth boundary. Although this underlying topological structure is smooth, the Riemannian metric tensor is only assumed to be bounded and measurable. This is known as a rough Riemannian manifold. For a large class of boundary condition…

Study spectral distribution of twisted Laplacian on high genus hyperbolic surfaces.

In this paper, by a new method we establish the Weyl-type asymptotic formula for the counting function of biharmonic Stekloff eigenvalues with Neumann boundary condition in a bounded domain of an $n$-dimensional Riemannian manifold.

Study nonlocal minimal surfaces for minimal surfaces in 3D manifolds.

The classical Weyl Law says that if $N_M(λ)$ denotes the number of eigenvalues of the Laplace operator on a $d$-dimensional compact manifold $M$ without a boundary that are less than or equal to $λ$, then $N_M(λ)=cλ^d+O(λ^{d-1}).$ In this paper, we show Duistermaat and Guillemin's result allows us to replace the $O(…

In this paper, we study the regularity of asymptotically hyperbolic metrics with Einstein condition near boundary and Weyl curvature smooth enough in arbitrary dimension. Following Michael Anderson's method, we show that $C^{m,α}$ conformally compact Riemannian metrics with Einstein equation vanishing to finite order n…

We discuss asymptotic behavior of the eigenvalue distribution of the differential form Laplacian on a Riemannian foliated manifold when the metric on the ambient manifold is blown up in directions normal to the leaves (in the adiabatic limit). Motivated by analogies with semiclassical spectral asymptotics, we use ideas…

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 …

We prove a Weyl Law for the phase transition spectrum based on the techniques of Liokumovich-Marques-Neves. As an application we give phase transition adaptations of the proofs of the density and equidistribution of minimal hypersufaces for generic metrics by Irie-Marques-Neves and Marques-Neves-Song, respectively. We …

Let (M, g) be a compact smooth Riemannian manifold. We obtain new off-diagonal estimates as λ 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 λ. A corollary is that, when rescaled around a non self-focal point, …

Symmetries of Einstein-Weyl manifolds can be extended from boundary surfaces.

Study gaps and clusters in eigenvalues of magnetic Laplacian on manifolds.

In this paper we consider the geometric behavior near infinity of some Einstein manifolds $(X^n, g)$ with Weyl curvature belonging to a certain $L^p$ space. Namely, we show that if $(X^n, g)$, $n \geq 7$, admits an essential set and has its Weyl curvature in $L^p$ for some $1<p<\frac{n-1}{2}$, then $(X^n, g)$ must be a…