arXiv subjects

Daniel Grieser

Publications and source records attributed to Daniel Grieser.

25 records · Page 2Linked to original sources

arXiv2007

Spectra of graph neighborhoods and scattering

Let $(G_ε)_{ε>0}$ be a family of '$ε$-thin' Riemannian manifolds modeled on a finite metric graph $G$, for example, the $ε$-neighborhood of an embedding of $G$ in some Euclidean space with straight edges. We study the asymptotic behavior of the spectrum of the Laplace-Beltrami operator on $G_ε$ as $ε\to 0$, for various boundary conditions. We obtain complete asymptotic expansions for the $k$th eigenvalue and the eigenfunctions, uniformly for $k\leq Cε^{-1}$, in terms of scattering data on a non-compact limit space. We then use this to determine the quantum graph which is to be regarded as the limit object, in a spectral sense, of the family $(G_ε)$. Our method is a direct construction of approximate eigenfunctions from the scattering and graph data, and use of a priori estimates to show that all eigenfunctions are obtained in this way.

math.SP↗

arXiv2007

Ueber Eigenwerte, Integrale und pi^2/6: Die Idee der Spurformel (On eigenvalues, integrals and pi^2/6: The idea of the trace formula)

This is an expository article that results from a talk given to second year students at Oldenburg university. The aim of the talk was to show what beautiful and unexpected results may be obtained if one plays with daring analogies in a way that is usually not done in undergraduate education (unfortunately): We start from the fact that the sum of diagonal entries of a symmetric matrix equals the sum of its eigenvalues. We then guess an analogous formula where the matrix is replaced by a function of two real variables and sums are replaced by integrals in a systematic way. We show that this is indeed a worthwhile process: In a special case it yields that the sum of inverse squares of the positive integers is pi^2/6. Finally, an outline of the proof of the guessed formula is given, and further applications, for example to the connection between billiards and the frequencies of a drum, are explained.

math.HO↗

arXiv2007

Asymptotics of eigenfunctions on plane domains

We consider a family of domains $(Ω_N)_{N>0}$ obtained by attaching an $N\times 1$ rectangle to a fixed set $Ω_0 = \{(x,y): 0<y<1, -ϕ(y)<x<0\}$, for a Lipschitz function $ϕ\geq 0$. We derive full asymptotic expansions, as $N\to\infty$, for the $m$th Dirichlet eigenvalue (for any fixed $m$) and for the associated eigenfunction on $Ω_N$. The second term involves a scattering phase arising in the Dirichlet problem on the infinite domain $Ω_\infty$. We determine the first variation of this scattering phase, with respect to $ϕ$, at $ϕ\equiv 0$. This is then used to prove sharpness of results, obtained previously by the same authors, about the location of extrema and nodal line of eigenfunctions on convex domains.

math.SP↗

arXiv2005

The first eigenvalue of the Laplacian, isoperimetric constants, and the Max Flow Min Cut Theorem

We show how 'test' vector fields may be used to give lower bounds for the Cheeger constant of a Euclidean domain (or Riemannian manifold with boundary), and hence for the lowest eigenvalue of the Dirichlet Laplacian on the domain. Also, we show that a continuous version of the classical Max Flow Min Cut Theorem for networks implies that Cheeger's constant may be obtained precisely from such vector fields. Finally, we apply these ideas to reprove a known lower bound for Cheeger's constant in terms of the inradius of a plane domain.

math.DG↗

arXiv2000

Basics of the b-calculus

R. B. Melrose's b-calculus provides a framework for dealing with problems of partial differential equations that arise in singular or degenerate geometric situations. This article is a somewhat informal short course introducing many of the basic ideas of this world, assuming little more than a basic analysis and manifold background. As examples, classical pseudodifferential operators on manifolds and b-pseudodifferential (also known as totally characteristic) operators on manifolds with boundary are discussed.

math.AP↗

arXiv2000

Singular Asymptotics Lemma and Push-Forward Theorem

The Singular Asymptotics Lemma by Brüning and Seeley and the Push-Forward Theorem by Melrose lie at the very heart of their respective approaches to singular analysis. We review both and show that they deal with the same basic problem, giving solutions that emphasize different aspects of it. This also points to a possible common extension.

math.DG↗

arXiv1999

Local Geometry of Singular Real Analytic Surfaces

Let V be a compact real analytic surface with isolated singularities embedded in $R^N$, and assume its smooth part is equipped with a Riemannian metric that is induced from some analytic Riemannian metric on $R^N$. We prove: 1. Each point of V has a neighborhood which is quasi-isometric (naturally and 'almost isometrically') to a union of metric cones and horns, glued at their tips. 2. A full asymptotic expansion, for any $p\in V$, of the length of $V\cap\{q:\dist(q,p)=r\}$ as r tends to zero. 3. A Gauss-Bonnet Theorem, saying that horns do not contribute an extra term, while cones contribute the leading coefficient in the length expansion of 2. 4. The $L^2$ Stokes Theorem, self-adjointness and discreteness of the Laplace-Beltrami operator on the smooth part, and a Gauss-Bonnet Theorem for the $L^2$ Euler characteristic. As a central tool we use resolution of singularities.

math.DG↗