- 01-318 Ricardo Weder
- Aharonov-Bohm Effect and Time-Dependent Inverse Scattering Theory
Sep 4, 01
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Abstract. We study the Aharonov-Bohm effect from the point of view of time-dependent inverse scattering theory. As this three dimensional problem is invariant under translations along the vertical axis, it reduces to a problem in $\ER^2$. We first consider an unshielded magnetic field that has a singular part produced by a tiny solenoid and a regular part. The wave function is zero at the location of the solenoid. We then consider the case where the singular part of the magnetic field is shielded inside a cylinder whose transversal section is compact set $K$, and there is also a regular magnetic field. In this case the magnetic field inside $K$ is quite general. Actually, the only condition is that the magnetic flux accross $ K$ has to be finite. Moreover, the wave function is defined in $\Omega:= \ER^2 \setminus K$ and it is zero on $\partial K$. We prove that in the unshielded case the scattering operator determines uniquely the regular magnetic field and that in the shielded case it determines uniquely the magnetic field in $\Omega$. Moreover, in the unshielded case the scattering operator determines the magnetic flux of the solenoid modulo 2 and in the shielded case it determines the magnetic flux across $K$ modulo 2. Our results follow from a reconstruction formula with error term.
This paper is a revised version of a previously posted manuscript.