Manuel Cañizares will defend his thesis on Wednesday, October 16th

• The defense will take place at Salón de Grados at the Faculty of Science and Technology of the Leioa Campus

Manuel Cañizares, studied Physics and Mathematics at Universidad de Sevilla, and then a master’s degree in Mathematical Physics at Universidad de Granada. His interests have always been in understanding physical phenomena with mathematical rigor. Cañizares works as a PhD student at the Basque Center for Applied Mathematics (BCAM) in the Harmonic Analysis and inverse problems research group (HA).

His thesis, titled Identifying quantum hamiltonians in the presence of electric interactions. An analytic approach is under the supervision of Pedro Caro (BCAM & Ikerbasque)

The defense is scheduled for Wednesday, October 16^th in Salón de Grados room at the Faculty of Science and Technology of the Leioa Campus at 15:00h.

On behalf of all members of BCAM, we would like to wish Manuel the best of luck in defending his thesis.

Abstract

In this thesis, we consider two inverse problems motivated by physical models related to quantum mechanics, particularly with the Schrödinger equation. These are the scattering problem with local near-field data and the problem of initial-to-final data in quantum mechanics. In both cases, we obtain an analytical uniqueness result related to the possibility of identifying time-independent electric potentials.

In the first problem, we demonstrate that potentials existing in functional spaces of low regularity can be identified by placing point sources and measuring the scattered wave at constant energy on small pieces of hypersurfaces near the support of the potential, in dimensions n≥3n \geq 3. To achieve this, we use harmonic analysis methods, domain perturbation techniques, and we prove a Runge approximation result.

In the second problem, we obtain a uniqueness result for bounded potentials with polynomial decay in dimensions n≥2n \geq 2. However, in this case, we do this by measuring the final state of a quantum system that interacts with the potential for every possible initial state.