Semiclassical study of eigenvalue crossings with applications to Physics .

• Grant number: 17/13865-0
• Support Opportunities: Scholarships in Brazil - Post-Doctoral
• Start date: September 01, 2017
• End date: August 31, 2019
• Field of knowledge: Physical Sciences and Mathematics - Physics
• Principal Investigator: João Carlos Alves Barata
• Grantee: Victor Bernardo Chabu
• Host Institution: Instituto de Física (IF). Universidade de São Paulo (USP). São Paulo , SP, Brazil

Abstract

In our PhD thesis we studied the propagation of Wigner measures linked to solutions of the Schrödinger equation with scalar potentials presenting conical singularities and showed that they are transported by two different Hamiltonian flows, one over the bundle cotangent to the singular set and the other elsewhere in the phase space, up to a transference phenomenon between these two regimes that may arise whenever trajectories in the outsider flow lead in or out the bundle. In the context of non-scalar potentials, which arise for instance in the statement of Schrödinger's equation for two levels molecules, the points where the potential's eigenvalues corresponding to the different energies cross each other form conical singularities exhibiting the same behavior as the trajectories in the scalar case, where in particular the Hamiltonian flow over the singularity's cotangent bundle would represent a resonant vibration mode allowing the molecule to change from one energy level to the other. In this project, we aim at pursuing the investigations we already started for the scalar case in order to describe this molecular phenomenon exactly. Moreover, in collaboration with Dr. Clotilde Fermanian and Dr. Fabricio Macià, we analyzed a Schrödinger-like equation pertinent to the semiclassical study of the dynamics of an electron in a crystal with impurities, showing that in the limit where the characteristic length of the crystal's lattice can be considered sufficiently small with respect to the variation of the exterior potential modeling the impurities, then this equation is approximated by an effective mass equation, or, more generally, that its solution decomposes in terms of Bloch modes, all of them satisfying effective mass equations specifically assigned to their Bloch energies. This was done under the hypothesis that each Bloch mode is isolated from all the others. Here, we intend to generalize this result to the case where different Bloch energies are allowed to cross, assuming only that they remain inside a Bloch band that be, itself, isolated.

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)

BARATA, J. C. A.; BRUM, M.; CHABU, V.; CORREA DA SILVA, R.. Pure and Mixed States. Brazilian Journal of Physics, v. 51, n. 2, SI, p. 244-262, APR 2021. (17/13865-0, 15/02975-4)