We consider a non compact, complete manifold {\bf{M}} of finite area with cuspidal ends. The generic cusp is isomorphic to ${\bf{X}}\times ]1,+\infty [$ with metric $ds^2=(h+dy^2)/y^{2δ}.$ {\bf{X}} is a compact manifold with nonzero first Betti number equipped with the metric $h.$ For a one-form $A$ on {\bf{M}} such that in each cusp $A$ is a non exact one-form on the boundary at infinity, we prove that the magnetic Laplacian $-Δ_A=(id+A)^\star (id+A)$ satisfies the Weyl asymptotic formula with sharp remainder. We deduce an upper bound for the counting function of the embedded eigenvalues of the Laplace-Beltrami operator $-Δ=-Δ_0.$
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 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.
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。
This paper proves sharp lower bounds on a resonance counting function for obstacle scattering in even-dimensional Euclidean space without a need for trapping assumptions. Similar lower bounds are proved for some other compactly supported perturbations of the Laplacian on even-dimensional Euclidean space, for example, for the Laplacian for certain metric perturbations. The proof uses a Poisson formula for resonances, complementary to one proved by Zworski in even dimensions.
The effective dynamics for a Bose-Einstein condensate in the regime of high dilution and subject to an external magnetic field is governed by a magnetic Gross-Pitaevskii equation. We elucidate the steps needed to adapt to the magnetic case the proof of the derivation of the Gross-Pitaevskii equation within the "projection counting" scheme.
We compute three-term semiclassical asymptotic expansions of counting functions and Riesz-means of the eigenvalues of the Laplacian on spheres and hemispheres, for both Dirichlet and Neumann boundary conditions. Specifically for Riesz-means we prove upper and lower bounds involving asymptotically sharp shift terms, and we extend them to domains of $\mathbb S^d$. We also prove a Berezin-Li-Yau inequality for domains contained in the hemisphere $\mathbb S^2_+$. Moreover, we consider polyharmonic operators for which we prove analogous results that highlight the role of dimension for Pólya-type in…
The asymptotic behaviour of the eigenvalue counting function of Laplacians on Hanoi attractors is determined. To this end, Dirichlet and resistance forms are constructed. Due to the non self-similarity of these sets, the classical construction of the Laplacian for p.c.f. self-similar fractals has to be modified by combining discrete and quantum graphs methods.
The aim of this article is to give a simple geometric condition that guarantees the existence of spectral gaps of the discrete Laplacian on periodic graphs. For proving this, we analyse the discrete magnetic Laplacian (DML) on the finite quotient and interpret the vector potential as a Floquet parameter. We develop a procedure of virtualising edges and vertices that produces matrices whose eigenvalues (written in ascending order and counting multiplicities) specify the bracketing intervals where the spectrum of the Laplacian is localised. We prove Higuchi-Shirai's conjecture for Z-periodic tre…
We discuss the application of Siegel Modular Forms to Black Hole entropy counting. The role of the Igusa cusp form $χ_{10}$ in the D1D5P system is well-known, and its transformation properties are what allows precision microstate counting in this case. We apply a similar method to extract the Fourier coefficients of other Siegel modular and paramodular forms, and we show that they could serve as candidates for other types of black holes. We investigate the growth of their coefficients, identifying the dominant contributions and the leading logarithmic corrections in various regimes. We also di…
This article studies potential theory and spectral analysis on compact metric spaces, which we refer to as fractal quantum graphs. These spaces can be represented as a (possibly infinite) union of 1-dimensional intervals and a totally disconnected (possibly uncountable) compact set, which roughly speaking represents the set of junction points. Classical quantum graphs and fractal spaces such as the Hanoi attractor are included among them. We begin with proving the existence of a resistance form on the Hanoi attractor, and go on to establish heat kernel estimates and upper and lower bounds on t…
