Andras Vasy
We show the exponential decay of eigenfunctions of second-order geometric many-body type Hamiltonians at non-threshold energies. Moreover, in the case of first order and small second order perturbations we show that there are no eigenfunctions with positive energy.
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We study the decay of eigenfunctions of the non self-adjoint matrix operator $\calH = (\begin{smallmatrix} -Δ+μ+U & W \W & Δ-μ-U \end{smallmatrix})$, for $μ>0$, corresponding to eigenvalues in the strip $-μ<\re E <μ$.
We consider non-self-adjoint electromagnetic Schrödinger operators on arbitrary open sets with complex scalar potentials whose real part is not necessarily bounded from below. Under a suitable sufficient condition on the electromagnetic potential, we introduce a Dirichlet realisation as a closed densely defined operator with non-empty resolvent set and show that the eigenfunctions corresponding to discrete eigenvalues satisfy an Agmon-type exponential decay.
This paper is devoted to the analysis of Steklov eigenvalues and Steklov eigenfunctions on a class of warped product Riemannian manifolds $(M,g)$ whose boundary $\partial M$ consists in two distinct connected components $Γ_0$ and $Γ_1$. First, we show that the Steklov eigenvalues can be divided into two families $(λ_m^\pm)_{m \geq 0}$ which satisfy accurate asymptotics as $m \to \infty$. Second, we consider the associated Steklov eigenfunctions which are the harmonic extensions of the boundary Dirichlet to Neumann eigenfunctions. In the case of symmetric warped product, we prove that the Stekl…
Exponential growth and decay estimates for eigenfunctions with complex eigenvalues
研究非自伴薛定谔算子具有复特征值的局部平方可积本征函数的指数增长和衰减估计，通过改编伊藤 - 斯基布斯特德的交换子方案，给出统一方法及额外估计。
In this paper we continue our study of the Laplacian on manifolds with axial analytic asymptotically cylindrical ends initiated in~arXiv:1003.2538. By using the complex scaling method and the Phragmén-Lindelöf principle we prove exponential decay of the eigenfunctions corresponding to the non-threshold eigenvalues of the Laplacian on functions. In the case of a manifold with (non-compact) boundary it is either the Dirichlet Laplacian or the Neumann Laplacian. We show that the rate of exponential decay of an eigenfunction is prescribed by the distance from the corresponding eigenvalue to the ne…
We study exponential decay rates of eigenfunctions of self-adjoint higher order elliptic operators on R^n. We are interested in decay rates as a function of direction. We show that the possible decay rates are to a large extent determined algebraically.
We study exponential decay of eigenfunctions of self-adjoint higher order elliptic operators on $\R^d$. We show that the possible critical decay rates are determined algebraically. In addition we show absence of super-exponentially decaying eigenfunctions and a refined exponential upper bound.
Exponential localization of 2d Magnetic Schrödinger eigenfunctions via Brownian flux
针对二维磁性 Schrödinger 算子的 Dirichlet 本征函数，本文借助 Feynman–Kac–Itô 公式分析热核，并用布朗路径相关的平均磁通量构造 Agmon 型距离，证明其向经典允许区域的指数局部化。
We prove rapid decay (even exponential decay under some stronger assumptions) of the eigenfunctions associated to discrete eigenvalues, for a class of self-adjoint operators in $L^2(\mathbb{R}^d)$ defined by ``magnetic'' pseudodifferential operators (studied in \cite{IMP1}). This class contains the relativistic Schrödinger operator with magnetic field.
Exponential Decays of Steklov Eigenfunctions for the Magnetic Laplacian
针对有界Lipschitz区域上带有限型磁场的磁薛定谔算子对应的Dirichlet-to-Neumann映射，证明场强足够大时其基态在磁场最大阶消失边界子集邻域外呈指数衰减。
Semiclassical localization of Schrödinger's eigenfunctions
本文通过改进的 Landis 猜想方法，建立了 Schrödinger 特征函数在闭合黎曼面上的指数界限，扩展了经典结果至有界势场的情况。
We study solutions of $-Δu + V u = λu$ on $\mathbb{R}^n$. Such solutions localize in the `allowed' region $\left\{x \in \mathbb{R}^n: V(x) \leq λ\right\}$ and decay exponentially in the `forbidden' region $\left\{x \in \mathbb{R}^n: V(x) > λ\right\}$. One way of making this precise is Agmon's inequality implying decay estimates in terms of the Agmon metric. We prove a complementary decay estimate in terms of harmonic measure which can improve on Agmon's estimate, connect the Agmon metric to decay of harmonic measure and prove a sharp pointwise Agmon estimate.
Eigenfunctions with double exponential rate of localization
本文在柱体和半柱体中分别构造了特征值问题和热方程的解，其衰减速度达到双指数级别，并回顾了Plis和Miller构造唯一延拓反例的经典思想。
Exponential decay for the Schr{ö}dinger equation on a dissipative waveguide
针对边界处处吸收却不满足几何控制条件的耗散波导，本文利用变量分离与横向算子特征函数的 Riesz 基性质，证明 Schrödinger 方程解呈指数衰减，并讨论负吸收指数情形。
We describe the propagation of singularities of tempered distributional generalized eigenfunctions of many-body Hamiltonians at non-threshold energies under the assumption that the inter-particle interactions are real-valued polyhomogeneous symbols of order -1 (e.g. Coulomb-type, but without the singularity at the origin). Here the term `singularity' refers to a microlocal description of the lack of decay at infinity. We use this result to describe the wave front relation of the S-matrices. We also analyze Lagrangian properties of this relation, which shows that the relation is not `too large'…
In this paper we describe the propagation of singularities of tempered distributional generalized eigenfunctions of many-body Hamiltonians under the assumption that no subsystem has a bound state and that the two-body interactions are real-valued polyhomogeneous symbols of order -1 (e.g. Coulomb-type with the singularity at the origin removed). Here the term 'singularity' provides a microlocal description of the lack of decay at infinity. We use this result to prove that the wave front relation of the free-to-free S-matrix (which, under our assumptions, is all of the S-matrix) is given by the …
Let $(M,g)$ be a compact, Riemannian manifold and $V \in C^{\infty}(M; \mathbb{R})$. Given a regular energy level $E > \min V$, we consider $L^2$-normalized eigenfunctions, $u_h,$ of the Schrodinger operator $P(h) = - h^2 Δ_g + V - E(h)$ with $P(h) u_h = 0$ and $E(h) = E + o(1)$ as $h \to 0^+.$ The well-known Agmon-Lithner estimates \cite{Hel} are exponential decay estimates (ie. upper bounds) for eigenfunctions in the forbidden region $\{ V>E \}.$ The decay rate is given in terms of the Agmon distance function $d_E$ associated with the degenerate Agmon metric $(V-E)_+ \, g$ with support in th…
We study a class of nonlinear eigenvalue problems of Schrödinger type, where the potential is singular on a set of points. Such problems are widely present in physics and chemistry, and their analysis is of both theoretical and practical interest. In particular, we study the regularity of the eigenfunctions of the operators considered, and we propose and analyze the approximation of the solution via an isotropically refined $hp$ discontinuous Galerkin (dG) method. We show that, for weighted analytic potentials and for up-to-quartic polynomial nonlinearities, the eigenfunctions belong to analyt…
Two-body Coulomb problem and $g^{(2)}$ algebra (once again about the Hydrogen atom)
本文以氢原子为例，揭示其对称性下的变量分离导致新的双变量正交多项式，并关联到$g^{(2)}$代数，应用于广义库仑问题与塞曼效应研究。
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