平面负位势相关的Schrdinger方程的非负广义解

NONNEGATIVE DISTRIBUTIONAL SOLUTIONS OF SCHRDINGER EQUATIONS WITH NEGATIVE PLANAR POTENTIALS

设β是复平面上圆盘 内的一个零容紧致集.考虑 上的定常Schrodinger方程(-△+μ)u=0,其中位势μ≤0是Kato类Radon测度,将方程在广义函数意义下的在 ,上取极限值0的非负连续解族记为μH+.对Ωαβ的Kerekjato-Stoilow意义下的理想边界β的任一点ζ,本文通过定义μH+→μH+的线性算子πζ,引入Martin函数Kζ,证明了μH+= 。

Let fif be a domain on the complex plan obtained by moving from a disc of radius a a totally disconnected compact set 3 of zero capacity. Consider a stationary Schrodinger equation (-A + p,)u = 0 on fi, where the potential j, < 0 is a signed Radon measure of Kato class. We Denote by H+ the class of all distributional solution of the equation which are nonnegative and continuous on fif vanishing on its boundary dtl = {z z = a}. For any f3, as an ideal boundary point in the Stoilow's sense, an operator : pH+ H+ is introduced, with which there is a Martin function K normalized by putting (ZD) = 1 for some z a- shown that H+ - Hp Pp, where