We obtain high-velocity estimates with error bounds for the scattering operator of the Schrödinger equation in three dimensions with electromagnetic potentials in the exterior of bounded obstacles that are handlebodies. A particular case is a finite number of tori. We prove our results with time-dependent methods. We consider high-velocity estimates where the direction of the velocity of the incoming electrons is kept fixed as its absolute value goes to infinity. In the case of one torus our results give a rigorous proof that quantum mechanics predicts the interference patterns observed in the fundamental experiments of Tonomura et al. that gave a conclusive evidence of the existence of the Aharonov-Bohm effect using a toroidal magnet. We give a method for the reconstruction of the flux of the magnetic field over a cross-section of the torus modulo 2π. Equivalently, we determine modulo 2π the difference in phase for two electrons that travel to infinity, when one goes inside the hole and the other outside it. For this purpose we only need the high-velocity limit of the scattering operator for one direction of the velocity of the incoming electrons. When there are several tori -or more generally handlebodiesthe information that we obtain in the fluxes, and on the difference of phases, depends on the relative position of the tori and on the direction of the velocities when we take the high-velocity limit of the incoming electrons. For some locations of the tori we can determine all the fluxes modulo 2π by taking the high-velocity limit in only one direction. We also give a method for the unique reconstruction of the electric potential and the magnetic field outside the handlebodies from the high-velocity limit of the scattering operator. 1The Aharonov-Bohm effect is a fundamental quantum mechanical phenomenon wherein charged particles, like electrons, are physically influenced, in the form of a phase shift, by the existence of magnetic fields in regions that are inaccessible to the particles. This genuinely quantum mechanical phenomenon was predicted by Aharonov and Bohm [3]. See also Ehrenberg and Siday [9]. This phenomenon has been extensively studied both, from the theoretical, and the experimental points of view. For a review of the literature see [29] and [30]. There has been a large controversy, involving over three hundred papers, concerning the existence of the Aharonov-Bohm effect. For a detailed discussion of this controversy see [30]. The issue was finally settled by the fundamental experiments of Tononura et al. [37,38], who used toroidal magnets to enclose a magnetic flux inside them. In remarkable experiments they were able to superimpose behind the magnet an electron beam that traveled inside the hole of the magnet with another electron beam that traveled outside the magnet, and they measured the phase shift produced by the magnetic flux enclosed in the magnet, giving a conclusive evidence of the existence of the Aharonov-Bohm effect.In this paper we give a rigorous mathematical analysis of this scat...
The Schrödinger equation on the half line is considered with a real-valued, integrable potential having a finite first moment. It is shown that the potential and the boundary conditions are uniquely determined by the data containing the discrete eigenvalues for a boundary condition at the origin, the continuous part of the spectral measure for that boundary condition, and a subset of the discrete eigenvalues for a different boundary condition. This result extends the celebrated two-spectrum uniqueness theorem of Borg and Marchenko to the case where there is also a continuous spectrum.
We prove that in multidimensional short-range potential scattering the high velocity limit of the scattering operator of an N-body system determines uniquely the potential. For a given long-range potential the short-range potential of the N-body system is uniquely determined by the high velocity limit of the modified Dollard scattering operator. Moreover, we prove that any one of the Dollard scattering operators determines uniquely the total potential. We obtain as well a reconstruction formula with an error term. Our simple proof uses a geometrical time-dependent method.
In this paper I prove a L p &L p estimate for the solutions to the one-dimensional Schro dinger equation with a potential in L 1 # where in the generic case #>3Â2 and in the exceptional case (i.e., when there is a half-bound state of zero energy) #>5Â2. I use this estimate to construct the scattering operator for the nonlinear Schro dinger equation with a potential. I prove moreover, that the low-energy limit of the scattering operator uniquely determines the potential and the coupling constant of the nonlinearity using a method that allows as well for the reconstruction of the potential and of the nonlinearity. Academic PressThe domain of H 0 , D(H 0 ), is the Sobolev space W 2 . The solution to (1.1) is given by e &itH 0 ,, where the strongly continuous unitary group e &itH 0 is defined by the functional calculus of self-adjoint operators. The kernel of Article ID jfan.1999.3507, available online at http:ÂÂwww.idealibrary.com on
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.
hi@scite.ai
10624 S. Eastern Ave., Ste. A-614
Henderson, NV 89052, USA
Copyright © 2024 scite LLC. All rights reserved.
Made with 💙 for researchers
Part of the Research Solutions Family.