= Zero modes of a positive transverse Pauli field
{title2=$\chi_k=e^{-ky-\int_0^y a(s)ds}$}
For $B=B(y)e_z$ with $B(y)\ge B_0>0$, take $a(y)=(e/\hbar)\int_0^yB(s)ds$. The spin-down transverse Hamiltonian at momentum label $-k$ is proportional to $M^\dagger M$, where $M=\partial_y+k+a(y)$. It is nonnegative, and $M\chi_k=0$ gives the displayed zero mode. Since $a'\ge eB_0/\hbar$, its exponent has a negative quadratic upper bound at both infinities, proving square integrability for every $k$. The continuously varying momentum label supplies an infinitely degenerate zero-energy space.
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