具有非符号定磁场的Pauli算子的加权$L_2$和零模的整函数
Entire functions in weighted $L_2$ and zero modes of the Pauli operator with non-signdefinite magnetic field
摘要：
对于实非符号定函数$B(z)$，$z\in \C$，研究了具有权$e^{\pm 2F}$平方可积的整解析函数空间的维数，其中函数$F(z)=F(x_1,x_2)$满足泊松方程$\D F=B$ .对于具有常数符号的函数$B$，答案是已知的。本文讨论了几类非符号定正齐次函数$B$，其中无穷维和零维都可以出现。在前一种情况下，我们提出了一种方法，构造整函数的规定行为在无穷远在不同的方向。这个题目与泡利算子的零能量子空间（零模）的维数问题密切相关。
英文摘要：
For a real non-signdefinite function $B(z)$, $z\in \C$, we investigate the dimension of the space of entire analytical functions square integrable with weight $e^{\pm 2F}$, where the function $F(z)=F(x_1,x_2)$ satisfies the Poisson equation $\D F=B$. The answer is known for the function $B$ with constant sign. We discuss some classes of non-signdefinite positively homogeneous functions $B$, where both infinite and zero dimension may occur. In the former case we present a method of constructing entire functions with prescribed behavior at infinity in different directions. The topic is closely related with the question of the dimension of the zero energy subspace (zero modes) for the Pauli operator.
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