Periodic Manifolds with Spectral Gaps

We investigate spectral properties of the Laplace operator on a class of non-compact Riemannian manifolds. For a given number $N$ we construct periodic (i.e. covering) manifolds such that the essential spectrum of the corresponding Laplacian has at least $N$ open gaps. We use two different methods. First, we construct a periodic manifold starting from an infinite number of copies of a compact manifold, connected by small cylinders. In the second construction we begin with a periodic manifold which will be conformally deformed. In both constructions, a decoupling of the different period cells is responsible for the gaps.Comment: 21 pages, 3 eps-figures, LaTe

Eigenvalues in Spectral Gaps of a Perturbed Periodic Manifold

We consider a non-compact Riemannian periodic manifold such that the corresponding Laplacian has a spectral gap. By continuously perturbing the periodic metric locally we can prove the existence of eigenvalues in a gap. A lower bound on the number of eigenvalue branches crossing a fixed level is established in terms of a discrete eigenvalue problem. Furthermore, we discuss examples of perturbations leading to infinitely many eigenvalue branches coming from above resp. finitely many branches coming from below.Comment: 30 pages, 3 eps-figures, LaTe

Spectral convergence of non-compact quasi-one-dimensional spaces

We consider a family of non-compact manifolds X_\eps (``graph-like manifolds'') approaching a metric graph $X_0$ and establish convergence results of the related natural operators, namely the (Neumann) Laplacian \laplacian {X_\eps} and the generalised Neumann (Kirchhoff) Laplacian \laplacian {X_0} on the metric graph. In particular, we show the norm convergence of the resolvents, spectral projections and eigenfunctions. As a consequence, the essential and the discrete spectrum converge as well. Neither the manifolds nor the metric graph need to be compact, we only need some natural uniformity assumptions. We provide examples of manifolds having spectral gaps in the essential spectrum, discrete eigenvalues in the gaps or even manifolds approaching a fractal spectrum. The convergence results will be given in a completely abstract setting dealing with operators acting in different spaces, applicable also in other geometric situations.Comment: some references added, still 36 pages, 4 figure

Operator estimates for the crushed ice problem

Let $\Delta_{\Omega_\varepsilon}$ be the Dirichlet Laplacian in the domain $\Omega_\varepsilon:=\Omega\setminus\left(\cup_i D_{i \varepsilon}\right)$. Here $\Omega\subset\mathbb{R}^n$ and $\{D_{i \varepsilon}\}_{i}$ is a family of tiny identical holes ("ice pieces") distributed periodically in $\mathbb{R}^n$ with period $\varepsilon$. We denote by $\mathrm{cap}(D_{i \varepsilon})$ the capacity of a single hole. It was known for a long time that $-\Delta_{\Omega_\varepsilon}$ converges to the operator $-\Delta_{\Omega}+q$ in strong resolvent sense provided the limit $q:=\lim_{\varepsilon\to 0} \mathrm{cap}(D_{i\varepsilon}) \varepsilon^{-n}$ exists and is finite. In the current contribution we improve this result deriving estimates for the rate of convergence in terms of operator norms. As an application, we establish the uniform convergence of the corresponding semi-groups and (for bounded $\Omega$) an estimate for the difference of the $k$-th eigenvalue of $-\Delta_{\Omega_\varepsilon}$ and $-\Delta_{\Omega_\varepsilon}+q$. Our proofs relies on an abstract scheme for studying the convergence of operators in varying Hilbert spaces developed previously by the second author.Comment: now 24 pages, 3 figures; some typos fixed and references adde