A note on Martin boundary of angular regions for Schrodinger equations
Toshimasa Tada · Journal of the Mathematical Society of Japan · 1989
We denote by $\Omega$ the punctured unit disk $0<|z|<1$ and consider the Martin compactification $\Omega_{P}^{*}$ ([4, p. 166]) of $\Omega$ with respect to a Schrodinger equation ( 1)with its potential $P$ on $\Omega$ .The potential $P$ on $\Omega$ is assumed to be nonnegative and locally Holder continuous on $0<|z|\leqq 1$ .We also consider the Martin com- pactification $A_{P}^{*}$ of an angular region $A$ with radius 1 and vertex at the origin $z=0$ with respect to (1).Let $\overline{\Omega}$ and $\overline{A}$ be the Euclidean closures of $\Omega$ and $A$ , respectively.One might ask the following QUESTION 1.Does $Af=\overline{A}$ for all angular regions $A$ imply $\Omega f=\overline{\Omega}P$ Here the equality $\Omega_{P}^{*}=\overline{\Omega}$ ( $A_{P}^{*}=\overline{A}$ , resp.) means that the identity mapping of $\Omega$ ( $A$ , resp.) can be extended to a homeomorphism of $\Omega_{P}^{*}$ ( $A_{P}^{*}$ , resp.) onto $\overline{\Omega}$ ( $\overline{A}$ , resp.).For a point $p$ in the Euclidean boundary $\partial\Omega$ ( $\partial A$ , resp.) of $\Omega$ ( $A$ , resp.), we denote by $\Omega_{P}^{*}(p)$ ( $A_{P}^{*}(p)$ , resp.) the set of all Martin boundary point $\zeta^{*}$ of $\Omega(A$ , resp.) for which there exists a sequence $\{\zeta_{n}\}_{1}^{\infty}$ in $\Omega$ ( $A$ , resp.) converging to $P$ with respect to the Euclidean topology and at the same time converging to $\zeta^{*}$ with respect to the Martin topology.We call $\Omega_{P}^{*}(p)$ ( $A_{P}^{*}(p)$ , resp.) the Martin boundary of $\Omega$ ( $A$ , resp.) over $p$ .We also denote by $\Omega_{P.1}^{*}(p)$ ( $A_{P,1}^{*}(p)$ , resp.) the set of Martin minimal boundary points over $p,$ $i.e$ .the subset of $\Omega_{P}^{*}(p)(A_{P}^{*}(p)$ , resp.) consisting of minimal points.In terms of $\Omega_{P,1}^{*}(0)$ and $A_{P.1}^{*}(0)$ ,Question 1 can be reformulated as QUESTION 2. Does $A_{P,1}^{*}(0)=$ { $one$ pojnt} for all angular regions $A$ imply $\Omega B_{1}(0)=\{one$ point $\}^{\mathcal{P}}$