相关论文: The Marked Length Spectrum Versus the Laplace Spec…

A general approach to proving that the length spectrum of a compact Riemannian manifold is an invariant of the Laplace spectrum comes from considering the wave trace, a spectrally determined tempered distribution. The Poisson relation…

Let $M$ be a $2m$-dimensional compact Riemannian manifold. We show that the spectrum of the Hodge Laplacian acting on $m$-forms does not determine whether the manifold has boundary, nor does it determine the lengths of the closed geodesics.…

In 2004, Sormani and Wei introduced the covering spectrum: a geometric invariant that isolates part of the length spectrum of a Riemannian manifold. In their paper they observed that certain Sunada isospectral manifolds share the same…

In this paper, we describe the spectrum properties of mixed operators, precisely the superposition of the classical Laplace operator and the fractional Laplace operator in the presence of mixed boundary conditions, that is \begin{equation}…

We prove a trace formula for three-dimensional spherically symmetric Riemannian manifolds with boundary which satisfy the Herglotz condition: The wave trace is singular precisely at the length spectrum of periodic broken rays. In…

In this paper are given explicit calculations of Laplace operator spectrum for smooth real/complex-valued functions on all connected compact simple rank three Lie groups with biinvariant Riemannian metric and established a connection of…

We define Dirichlet type series associated with homology length spectra of Riemannian, or Finsler, manifolds, or polyhedra, and investigate some of their analytical properties. As a consequence we obtain an inequality analogous to Gromov's…

Given a manifold (or, more generally, a developable orbifold) $M_0$ and two closed Riemannian manifolds $M_1$ and $M_2$ with a finite covering map to $M_0$, we give a spectral characterisation of when they are equivalent Riemannian covers…

We construct Riemannian manifolds with singular continuous spectrum embedded in the absolutely continuous spectrum of the Laplacian. Our manifolds are asymptotically hyperbolic with sharp curvature bounds.

We construct a Laplace isospectral deformation of metrics on an orbifold quotient of a nilmanifold. Each orbifold in the deformation contains singular points with order two isotropy. Isospectrality is obtained by modifying a generalization…

In this paper, we show that simple, thick negatively curved two-dimensional P-manifolds, a large class of surface amalgams, are marked length spectrum rigid. That is, if two piecewise negatively curved Riemannian metrics (satisfying certain…

Let $(M,g)$ be a non-compact riemannian $n$-manifold with bounded geometry at order $k\geq\frac{n}{2}$. We show that if the spectrum of the Laplacian starts with $q+1$ discrete eigenvalues isolated from the essential spectrum, and if the…

We consider the spectrum of the Laplace operator acting on $\mathcal{L}^p$ over a conformally compact manifold for $1 \leq p \leq \infty$. We prove that for $p \neq 2$ this spectrum always contains an open region of the complex plane. We…

We compare the marked length spectra of isometric actions of groups with non-positively curved features. Inspired by the recent works of Butt we study approximate versions of marked length spectrum rigidity. We show that for pairs of…

We define a new spectrum for compact length spaces and Riemannian manifolds called the "covering spectrum" which roughly measures the size of the one dimensional holes in the space. More specifically, the covering spectrum is a set of real…

We consider open manifolds which are interiors of a compact manifold with boundary, and Riemannian metrics asymptotic to a conformally cylindrical metric near the boundary. We show that the essential spectrum of the Laplace operator on…

We study spectral properties of the standard (also called Kirchhoff) Laplacian and the anti-standard (or anti-Kirchhoff) Laplacian on a finite, compact metric graph. We show that the positive eigenvalues of these two operators coincide…

In this paper we consider a family of Riemannian manifolds, not necessarily complete, with curvature conditions in a neighborhood of a ray. Under these conditions we obtain that the essential spectrum of the Laplacian contains an interval.…

We consider a closed Riemannian manifold $M$ of negative curvature and dimension at least 3 with marked length spectrum sufficiently close (multiplicatively) to that of a locally symmetric space $N$. Using the methods of Hamenst\"adt, we…

In this article we prove that the spectrum of the Laplacian on $k$-forms over a noncompact flat manifold is always a connected closed interval of the nonnegative real line. The proof is based on a detailed decomposition of the structure of…