03-379 Yafaev D.
There is no Aharonov-Bohm effect in dimension three (50K, LATeX 2e) Aug 22, 03
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Abstract. Consider the scattering amplitude $s(\omega,\omega^\prime;\lambda)$, $\omega,\omega^\prime\in{\Bbb S}^2$, $\lambda>0$, corresponding to an arbitrary three-dimensional short-range magnetic field $B(x)$. The magnetic potential $A^{(tr)}(x)$ such that ${\rm curl}\,A^{(tr)}(x)=B(x)$ and $<A^{(tr)}(x),x>=0$ decays at infinity as $|x|^{-1}$ only. Nevertheless, we show that the structure of $s(\omega,\omega^\prime;\lambda)$ is the same as for short-range magnetic potentials. In particular, the leading diagonal singularity $s_0(\omega,\omega^\prime)$ of $s(\omega,\omega^\prime;\lambda)$ is the Dirac function. Thus, up to the diagonal Dirac function, the scattering amplitude has only a weak singularity in the forward direction and hence scattering is essentially of short-range nature. This is qualitatively different from the two-dimensional case where $s_0(\omega,\omega^\prime)$ is a linear combination of the Dirac function and of a singular denominator, that is the Aharonov-Bohm effect occurs. Our approach relies on a construction of a special gauge, adapted to a given magnetic field $B(x)$, such that the corresponding magnetic potential $A (x)$ is also short-range.

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