On the zero modes of Pauli operators

Abstract

Two results are proved for $\mathrm{nul}{\mathbb{P}}_A$, the dimension of the kernel of the Pauli operator $\mathbb{P}_A = \bigl\{\bbfσ \cdotp \bigl(\frac{1}{i} \bbf{\nabla} + \vec{A} \bigr) \bigr\} ^2 $ in $[L^2({\mathbb{R}}^3){]}^2$: (i) for $|\vec{B}|\in L^{3/2}({\mathbb{R}}^3),$ where $\vec{B}=\mathrm{curl}\vec{A}$ is the magnetic field, $\mathrm{nul}{\mathbb{P}}_{tA}=0$ except for a finite number of values of $t$ in any compact subset of $(0,\infty )$; (ii) $\{\vec{B}:\mathrm{nul}{\mathbb{P}}_A=0,|\vec{B}|\in L^{3/2}({\mathbb{R}}^3)\}$ contains an open dense subset of $[L^{3/2}({\mathbb{R}}^3){]}^3$.

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