We consider an open manifold which is the interior of a compact manifold with boundary. Assuming gauge invariance, we classify magnetic fields with compact support into being trapping or non-trapping. We study spectral properties of the associated magnetic Laplacian for a class of Riemannian metrics which includes complete hyperbolic metrics of finite volume. When $B$ is non-trapping, the magnetic Laplacian has nonempty essential spectrum. Using Mourre theory, we show the absence of singular continuous spectrum and the local finiteness of the point spectrum. When $B$ is trapping, the spectrum is discrete and obeys the Weyl law. The existence of trapping magnetic fields with compact support depends on cohomological conditions, indicating a new and very strong long-range effect.
In the non-gauge invariant case, we exhibit a strong Aharonov-Bohm effect. On hyperbolic surfaces with at least two cusps, we show that the magnetic Laplacian associated to every magnetic field with compact support has purely discrete spectrum for some choices of the vector potential, while other choices lead to a situation of limit absorption principle.
We also study perturbations of the metric. We show that in the Mourre theory it is not necessary to require a decay of the derivatives of the perturbation. This very singular perturbation is then brought closer to the perturbation of a potential.
The spectrum of Schrödinger operators and Hodge Laplacians on conformally cusp manifolds
本文描述了共形尖点黎曼流形上的$k$ -形式拉普拉斯算子的谱.当边界的$k$和$k-1$ de Rham上同调群消失时，本质谱精确地消失.我们给出了纯离散情形下特征值计数函数的Weyl型渐近性。在另一种情况下，我们用正交换子方法分析了本质谱，并建立了极限吸收原理。这意味着没有奇异谱的一个广泛的类的度量。我们还展示了一类势$V$，使得薛定谔算子具有紧预解式，尽管$V$在大部分无穷远处趋于$-\infty$。本文纠正了文献中关于有限体积双曲流形上的拉普拉斯算子的本质谱的一个说法，并提出了关于四维中尖点为有理同调球面的这种流形的存在性的一个猜想.
We introduce the notion of discrete cusp for a weighted graph. In this context, we provethat the form-domain of the magnetic Laplacian and that of thenon-magnetic Laplacian can be different. We establish the emptiness of the essential spectrum and compute theasymptotic of eigenvalues for the magnetic Laplacian.
The spectrum of magnetic Schrödinger operators and $k$-form Laplacians on conformally cusp manifolds
我们考虑具有边界的紧致带边流形内部的开流形，并且黎曼度量在边界附近渐近于共形圆柱度量。我们表明，在磁场的存在下，不定义一个完整的相对上同调类的功能的拉普拉斯算子的本质谱消失。由此可见，即使在强磁场的作用下，本质谱也不稳定。我们也处理磁场经营者扰动电场。在同一背景下，我们描述的本质谱的$k$形式的拉普拉斯算子。当边界的$k$和$k-1$ de Rham上同调群消失时，这一点恰好消失。在所有纯点谱情形下，我们给出了特征值计数函数的Weyl型渐近性。在其他情况下，我们描述本质谱。
We study perturbations of the discrete Laplacian associated to discrete analogs of cusps and funnels. We perturb the metric and the potential in a long-range way. We establish a propagation estimate and a Limiting Absorption Principle away from the possible embedded eigenvalues. The approach is based on a positive commutator technique.
We consider the Robin Laplacian in the domains $Ω$ and $Ω^\varepsilon$, $\varepsilon >0$, with sharp and blunted cusps, respectively. Assuming that the Robin coefficient $a$ is large enough, the spectrum of the problem in $Ω$ is known to be residual and to cover the whole complex plane, but on the contrary, the spectrum in the Lipschitz domain $Ω^\varepsilon$ is discrete. However, our results reveal the strange behavior of the discrete spectrum as the blunting parameter $\varepsilon$ tends to 0: we construct asymptotic forms of the eigenvalues and detect families of "hardly movable" and "plumm…
Let $X$ be a two-dimensional smooth manifold with boundary $S^{1}$ and $Y=[1,\infty)\times S^{1}$. We consider a family of complete surfaces arising by endowing $X\cup_{S^{1}}Y$ with a parameter dependent Riemannian metric, such that the restriction of the metric to $Y$ converges to the hyperbolic metric as a limit with respect to the parameter. We describe the associated spectral and scattering theory of the Laplacian for such a surface. We further show that on $Y$ the zero $S^{1}$-Fourier coefficient of the generalized eigenfunction of this Laplacian, as a family with respect to the paramete…
Manifolds with fibered cusps are a class of complete noncompact Riemannian manifolds including all locally symmetric spaces of rank one. We study the spectrum of the Hodge Laplacian with coefficients in a flat bundle on a closed manifold undergoing degeneration to a manifold with fibered cusps. We obtain precise asymptotics for the resolvent, the heat kernel, and the determinant of the Laplacian. Using these asymptotics we obtain a topological description of the analytic torsion on a manifold with fibered cusps in terms of the R-torsion of the underlying manifold with boundary.
We show that for any $ε>0$, $α\in[0,\frac{1}{2})$, as $g\to\infty$ a generic finite-area genus g hyperbolic surface with $n=O\left(g^α\right)$ cusps, sampled with probability arising from the Weil-Petersson metric on moduli space, has no non-zero eigenvalue of the Laplacian below $\frac{1}{4}-\left(\frac{2α+1}{4}\right)^{2}-ε$. For $α=0$ this gives a spectral gap of size $\frac{3}{16}-ε$ and for any $α<\frac{1}{2}$ gives a uniform spectral gap of explicit size.
We establish a Cheeger-Muller theorem for unimodular representations satisfying a Witt condition on a noncompact manifold with cusps. This class of spaces includes all non-compact hyperbolic spaces of finite volume, but we do not assume that the metric has constant curvature nor that the link of the cusp is a torus. We use renormalized traces in the sense of Melrose to define the analytic torsion and we relate it to the intersection R-torsion of Dar of the natural compactification to a stratified space. Our proof relies on our recent work on the behavior of the Hodge Laplacian spectrum on a cl…
The paper presents evidence that Riemann's xi function evaluated at 2 sqrt(E) could be the characteristic function P(E) for the magnetic Laplacian minus 85/16 on a surface of curvature -1 with magnetic field 9/4, a cusp of width 1, a DIrichlet condition at a point, and other conditions not yet determined.
We generalize work of Deligne and Gillet-Soulé on a Riemann-Roch type isometry, to the case of the trivial sheaf on cusp compactifications of Riemann surfaces $Γ\backslash\mathbb{H}$, for $Γ\subset PSL_{2}(\mathbb{R})$ a fuchsian group of the first kind, equipped with the Poincaré metric. This metric is singular at cusps and elliptic fixed points, and the results of Deligne and Gillet-Soulé do not apply to this setting. Our theorem relates the determinant of cohomology of the trivial sheaf, with an explicit Quillen type metric in terms of the Selberg zeta function of $Γ$, to a metrized version…
