Jialun Li
Let M be a geometrically finite rank one locally symmetric manifolds. We prove that the spectrum of the Laplace operator on M is finite in a small interval which is optimal.
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We prove conditions on potentials which imply that the sum of the negative eigenvalues of the Schroeodinger operator is finite. We use a method for bounding eigenvalues based on estimates of the Hilbert-Schmidt norm of semigroup differences and on complex analysis
We present a new oscillation criterion to determine whether the number of eigenvalues below the essential spectrum of a given Jacobi operator is finite or not. As an application we show that Kenser's criterion for Jacobi operators follows as a special case.
Let G=SO(n,1) and Gamma a geometrically finite Zariski dense subgroup of G which is contained in an arithmetic subgroup of G. Denoting by Gamma(q) the principal congruence subgroup of Gamma of level q, and fixing a positive number λ_0 strictly smaller than (n-1)^2/4, we show that, as q tends to infinity along primes, the number of Laplacian eigenvalues of the congruence cover Gamma(q)\ H^n smaller than lambda_0 is at most of order [Gamma:Gamma(q)]^c for some c=c(λ_0)>0.
Small eigenvalues of Schrödinger operators over geometrically finite manifolds
本文研究了几何有限流形上Riemannian向量丛的Schrödinger算子，估计了其小特征值的数量。
The Steklov spectrum of convex polygonal domains I: spectral finiteness
本文研究凸多边形的 Steklov 反问题，证明几乎所有凸多边形至多有有限多个等谱但不全等的区域，并给出数量上界及特征值等周界。
The goal of this paper is to derive estimates of eigenvalue moments for Dirichlet Laplacians and Schrödinger operators in regions having infinite cusps which are geometrically nontrivial being either curved or twisted; we are going to show how those geometric properties enter the eigenvalue bounds. The obtained inequalities reflect the essentially one-dimensional character of the cusps and we give an example showing that in an intermediate energy region they can be much stronger than the usual semiclassical bounds.
In this article, we prove the finiteness of the number of eigenvalues for a class of Schrödinger operators $H = -Δ+ V(x)$ with a complex-valued potential $V(x)$ on $\bR^n$, $n \ge 2$. If $\Im V$ is sufficiently small, $\Im V \le 0$ and $\Im V \neq 0$, we show that $N(V) = N(\Re V)+ k$, where $k$ is the multiplicity of the zero resonance of the selfadjoint operator $-Δ+ \Re V$ and $N(W)$ the number of eigenvalues of $-Δ+ W$, counted according to their algebraic multiplicity.
A lattice model of radiative decay (so-called spin-boson model) of a two level atom and at most two photons is considered. The location of the essential spectrum is described. For any coupling constant the finiteness of the number of eigenvalues below the bottom of its essential spectrum is proved. The results are obtained by considering a more general model $H$ for which the lower bound of its essential spectrum is estimated. Conditions which guarantee the finiteness of the number of eigenvalues of $H,$ below the bottom of its essential spectrum are found. It is shown that the discrete spectr…
We consider the discrete spectrum of the two-dimensional Hamiltonian $H=H_0+V$, where $H_0$ is a Schrödinger operator with a non-constant magnetic field $B$ that depends only on one of the spatial variables, and $V$ is an electric potential that decays at infinity. We study the accumulation rate of the eigenvalues of H in the gaps of its essential spectrum. First, under some general conditions on $B$ and $V$, we introduce effective Hamiltonians that govern the main asymptotic term of the eigenvalue counting function. Further, we use the effective Hamiltonians to find the asymptotic behavior of…
Absence of principal eigenvalues for higher rank locally symmetric spaces
本文将Patterson关于无限体积几何有限双曲曲面正Laplace-Beltrami算子无≥1/4的L²特征值的结果，推广到高秩局部对称空间的交换微分算子代数联合L²特征值情形，给出保证主特征值不存在的动力学假设，并指出Anosov子群的非紧商满足该假设。
Finiteness and exponential growth of full graph $p$-spectra
本文证明有限图 $p$-拉普拉斯算子的完全谱有限，解决 Amghibech 猜想，并给出谱基数与拓扑量的指数界，揭示 $p=2$ 处的转变及指数增长率为 3 的极值现象。
We study the spectral properties of discrete Schrödinger operator $$ \widehat H_μ=\widehat H_0 + μ\widehat{V},\qquad μ\ge0, $$ associated to a one-particle system in $d$-dimensional lattice $\mathbb{Z}^d, $ $d=1,2,$ where the non-perturbed operator $\hat H_0$ is a self-adjoint Laurent-Toeplitz-type operator generated by $\hat e:\mathbb{Z}^d\to\mathbb{C}$ and the potential $\hat V$ is the multiplication operator by $\hat v:\mathbb{Z}^d\to\mathbb{R}.$ Under certain regularity assumption on $\hat e$ and a decay assumption on $\hat v$, we establish the existence or non-existence and also the finit…
Principal $p-$frequency estimates on non-compact manifolds with negative Ricci curvature
研究非紧致负曲率流形上有界域上的主 $p$-频数下界，通过 Ricci 曲率条件得出明确的下限表达式，并证明该估计是最佳的。
Schrödinger Operators, Integral Curvature, and the Euler Characteristic of Riemannian Manifolds
本文通过薛定谔型算子与积分曲率界的关系，揭示了黎曼流形欧拉特征与曲率条件之间的拓扑联系，并给出了相关特征值下界及Yau问题的解答。
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