Confirmation of the Aharonov-Bohm effect without interaction with the solenoid border

• Grant number: 12/21480-8
• Support Opportunities: Scholarships in Brazil - Doctorate
• Start date: April 01, 2013
• End date: May 31, 2016
• Field of knowledge: Physical Sciences and Mathematics - Mathematics - Applied Mathematics
• Principal Investigator: Cesar Rogerio de Oliveira
• Grantee: Renan Gambale Romano
• Host Institution: Centro de Ciências Exatas e de Tecnologia (CCET). Universidade Federal de São Carlos (UFSCAR). São Carlos , SP, Brazil

Abstract

In his Master's Thesis, the candidate Renan has studied a situation of the Aharonov-Bohm effect (AB) with a scalar potential V(x) that diverges at the solenoid border. Based on recent results in the literature, it is shown that if such divergence is fast enough then the AB operator is essentially self-adjoint; this interesting situation is understood as no interaction with the solenoid border.In this project it is proposed a continuation of the researches associated with such model, now restricted to the plane. The main question will be to confirm, in fact, the presence of the AB effect in this situation without contact with the solenoid. First, by analysing in which cases the Hamiltonian operators H(A), with different magnetic potentials A, are unitarily equivalent. Second, by investigating the possible relations between the first eigenvalues p(A1) and p(A2) of H(A1) and H(A2), respectively; based on similar studies in the literature, in particular in the AB case with Dirichlet boundary conditions, we conjecture that p(A1)=p(A2) if, and only if, H(A1) and H(A2) are unitarily equivalent. To prove this is the second goal of this project.Finally, it is of interest to estimate the difference p(0)-p(rA) for r converging to zero. It would be a quantitative characterization of how the magnetic potential influences the eigenvalues. There are at least two possible techniques to be applied; one semiclassical due to Helffer and Sjöstrand, and another developed by Malliavin (and applied by Shigekawa) related to the heat kernel exp(-tH(A)) as t goes to infinity. The choice of the appropriate technique will be another step in the development of this doctorate. (AU)

Scientific publications

(References retrieved automatically from Web of Science and SciELO through information on FAPESP grants and their corresponding numbers as mentioned in the publications by the authors)

DE OLIVEIRA, CESAR R.; ROMANO, RENAN G.. Aharonov-Bohm effect without contact with the solenoid. Journal of Mathematical Physics, v. 58, n. 10, OCT 2017. (12/21480-8)