相关论文: Generating spectral gaps by geometry

We establish an $L^2$-Gamma index theorem for the Dirac operator on a globally hyperbolic manifold $M$ with Cauchy hypersurface $\Sigma$ being a Galois covering of a compact smooth manifold with Galois group $\Gamma$. Our argument rewrites…

Many "real" inverse spectral data for periodic finite-gap operators (consisting of Riemann Surface with marked "infinite point", local parameter and divisors of poles) lead to operators with real but singular coefficients. These operators…

We consider the Dirichlet Laplacian in a straight planar strip perturbed by a bounded periodic symmetric operator. We prove the classical Bethe-Sommerfeld conjecture for this operator, namely, that this operator has finitely many gaps in…

Suppose a group $G$ acts properly on a simplicial complex $\Gamma$. Let $l$ be the number of $G$-invariant vertices and $p_1, p_2, ... p_m$ be the sizes of the $G$-orbits having size greater than 1. Then $\Gamma$ must be a subcomplex of…

Let $\Gamma$ be an arbitrary $\mathbb{Z}^n$-periodic metric graph, which does not coincide with a line. We consider the Hamiltonian $\mathcal{H}_\varepsilon$ on $\Gamma$ with the action $-\varepsilon^{-1}{\mathrm{d}^2/\mathrm{d} x^2}$ on…

A compact topological space X is spectral if it is sober (i.e., every irreducible closed set is the closure of a unique singleton) and the compact open subsets of X form a basis of the topology of X, closed under finite intersections.…

We construct algebras of pseudodifferential operators on a continuous family groupoid G that are closed under holomorphic functional calculus, contain the algebra of all pseudodifferential operators of order 0 on G as a dense subalgebra,…

Let $G$ be a connected, simply connected nilpotent Lie group, identified with a real algebraic subgroup of $\mathrm{UT}(n,\mathbb{R})$, and let $\Gamma$ be a lattice in $G$, with $\pi:G\to G/\Gamma$ the quotient map. For a semi-algebraic…

For non-self-adjoint almost-periodic Schr\"odinger operators, a criterion is given to guarantee that they have both the same spectrum and same Lyapunov exponents with the discrete free Laplacian. As a byproduct, we show that the…

From a graph $G$ with constant valency $v$ and a (non-compact) manifold $C$ with $v$ boundary components, we build a $G$-periodic manifold $M$. This process gives a class of topologically infinite manifolds which generalizes periodic…

Let $ \{A^i,E^i\} $ be the elliptic complex on a $ n $-dimensional smooth closed Riemannian manifold $X$ with the first order differential operators $ A^i $ and smooth vector bundles $ E^i $ over $X$. We consider nonlinear operator…

We study the asymptotic behaviour of the spectral gap of Schr\"odinger operators in two and higher dimensions and in a limit where the volume of the domain tends to infinity. Depending on properties of the underlying potential, we will find…

We introduce a permutation model for random degree $n$ covers $X_{n}$ of a non-elementary convex-cocompact hyperbolic surface $X=\Gamma\backslash\mathbb{H}$. Let $\delta$ be the Hausdorff dimension of the limit set of $\Gamma$. We say that…

On a compact Riemannian manifold with boundary, we prove a spectral inequality for the bi-Laplace operator in the case of so-called "clamped" boundary conditions , that is, homogeneous Dirichlet and Neumann conditions simultaneously. We…

For the one-dimensional Schr\"odinger equation, some real intervals with no eigenvalues (the spectral gaps) may be obtained rather systematically with a method proposed by H. Giacomini and A. Mouchet in 2007. The present article provides…

We study the effect of non-negative potentials on the spectral gap of one-dimensional Schr\"odinger operators in the limit of large intervals. In particular, we derive upper and lower bounds on the gap for different classes of potentials…

This work establishes rigorous mathematical foundations connecting spectral graph theory, algebraic geometry, and string theory. We construct a canonical mapping whereby any finite graph \(G\) defines a compact Riemann surface \(X_{G}\)…

We study finite G-sets and their tensor product with Riemannian manifolds, and obtain results on isospectral quotients and covers. In particular, we show the following: if M is a compact connected Riemannian manifold (or orbifold) whose…

Let $(M,g)$ be a complete non-compact Riemannian manifold. We consider operators of the form $\Delta_g + V$, where $\Delta_g$ is the non-negative Laplacian associated with the metric $g$, and $V$ a locally integrable function. Let $\rho :…

We obtain an essential spectral gap for a convex co-compact hyperbolic surface $M=\Gamma\backslash\mathbb H^2$ which depends only on the dimension $\delta$ of the limit set. More precisely, we show that when $\delta>0$ there exists…