Zero modes of a positive transverse Pauli field (${\chi }_k=e^{-ky-{\int }_0^{y}a(s)ds}$)

For $B=B(y)e_z$ with $B(y)\ge B_0>0$, take $a(y)=(e/\hslash ){\int }_0^{y}B(s)ds$. The spin-down transverse Hamiltonian at momentum label $-k$ is proportional to $M^{†}M$, where $M={\partial }_y+k+a(y)$. It is nonnegative, and $M{\chi }_k=0$ gives the displayed zero mode. Since $a^{\prime }\ge e{B}_0/\hslash$, 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.