Sharp spectral estimates in domains of infinite volume

Sharp Tunneling Estimates for a Double-Well Model in Infinite Dimension

We consider the stochastic quantization of a quartic double-well energy functional in the semiclassical regime and derive optimal asymptotics for the exponentially small splitting of the ground state energy. Our result provides an infinite-dimensional version of some sharp tunneling estimates known in finite dimensions for semiclassical Witten Laplacians in degree zero. From a stochastic point of view it proves that the $L^2$ spectral gap of the stochastic one-dimensional Allen-Cahn equation in finite volume satisifies a Kramers-type formula in the limit of vanishing noise. We work with finite…

Some upper bounds on the number of resonances for manifolds with infinite cylindrical ends

We prove some sharp upper bounds on the number of resonances associated with the Laplacian, or Laplacian plus potential, on a manifold with infinite cylidrical ends.

Hyperbolic Eisenstein series for geometrically finite hyperbolic surfaces of infinite volume

For a geometrically finite hyperbolic surface of infinite volume we write down the spectral decomposition for the Laplacian on 1-forms, generalize the Kudla and Millson's construction of hyperbolic Eisenstein series and other related results.

L-functions and sharp resonances of infinite index congruence subgroups of $SL_2(\mathbb{Z})$

For convex co-compact subgroups of SL2(Z) we consider the "congruence subgroups" for p prime. We prove a factorization formula for the Selberg zeta function in term of L-functions related to irreducible representations of the Galois group SL2(Fp) of the covering, together with a priori bounds and analytic continuation. We use this factorization property combined with an averaging technique over representations to prove a new existence result of non-trivial resonances in an effective low frequency strip.

从局部块计算无限体积算子的谱和伪谱

Computing the spectrum and pseudospectrum of infinite-volume operators from local patches

该研究提出仅用有限局部块数据通过双侧误差控制近似正规离散短程无限体积算子的谱，证明其可计算性并提供算法，首次实现谱不存在性的计算确定，还扩展至非正规算子的ε-伪谱。

前 $N$ 个非零 Steklov 特征值的尖锐固定体积乘积不等式

A sharp fixed-volume product inequality for the first $N$ nonzero Steklov eigenvalues

针对 $\mathbb R^N$ 中有界 Lipschitz 域，证明了前 $N$ 个非零 Steklov 特征值的乘积不超过单位球体积与域体积之比，将凸域结果推广至任意有界 Lipschitz 域。