Subcriticality, positivity, and gaugeability of the Schrödinger operator
Subcriticality, positivity, and gaugeability of the Schrödinger operator
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薛定谔算子的次临界性、正性和可测性
DOI：
10.1090/s0273-0979-1990-15965-8
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发表时间：
1990
影响因子：
1.3
通讯作者：
Z. Zhao
中科院分区：
文献类型：
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作者：
Z. Zhao
我们从以下三个方面研究了R（d>3）中薛定谔算子H：= -（A/2）+ V > 0的性质：（I）次临界性：直觉上，如果H > 0是次临界的，那么应该可以通过小扰动来扰动H，并且仍然保持它的非负性。更确切地说，我们有以下断言：（a）对于任何q e Bc（Bc表示具有紧支集的有界Borel函数类），存在e > 0使得-（A/2）+ V + eq > 0。(b)存在一个函数q e Bc，q < 0且q ^ 0 a.e.使得-（A/2）+ V + q > 0。次临界状态还有另外两种定义：（c）（B）。Simon [7]）存在fi > 0使得-（A/2）+（l+fi）V>0。(d)（M. Murata [6]）对于f（？），存在一个正的绿色函数G（-，-）. (II)强正性：（e）存在Hu = 0的正解u > 0，极限为：lim，^ u（x）> 0。(f)存在Hu = 0的解u，且c > u > c > 0。(g)存在Hu = 0的解u，且u > c > 0。(III)可度量性：设{Xt：t > 0}是R中的布朗运动，设E表示从xeR开始的布朗路径上的期望。设u0（x）= ε *[exp（/0°° V（Xs）ds）]。(h)w0（.x）^ oo在R d . (i)u 0（x）在i中有界？.
We investigate properties of the Schrödinger operator H := -(A/2) + V > 0 in R(d>3) in the following three aspects: (I) Subcriticality: Intuitively, the idea is that if H > 0 is subcritical, then it should be possible to perturb H by small perturbations and still keep its nonnegativity. More precisely, we have the following assertions: (a) For any q e Bc (Bc denotes the class of bounded Borel functions with compact support), there exists an e > 0 such that -(A/2) + V + eq > 0. (b) There exists a function q e Bc, q < 0 and q ^ 0 a.e. such that -(A/2) + V + q > 0. There have been two other definitions of subcriticality: (c) (B. Simon [7]) There exists fi > 0 such that -(A/2) + (l+fi)V>0. (d) (M. Murata [6]) There exists a positive Green function G(-,-) for /ƒ. (II) Strong Positivity: (e) There exists a positive solution u > 0 of Hu = 0 with the limit: lim, , ^ u(x) > 0. (f) There exists a solution u of Hu = 0 with c > u > c > 0. (g) There exists a solution u of Hu = 0 with u > c > 0. (III) Gaugeability: Let {Xt : t > 0} be the Brownian motion in R and let E denote the expectation over the Brownian paths starting from x e R . Put u0(x) := £*[exp(/0°° V(Xs) ds)]. (h) w0(.x) ^ oo in R d . (i) u0(x) is bounded in i? .