I graduated in Physics in 2008 and obtained a MSc degree in Theoretical Physics in 2009. Due to my personal interests in a better understanding of the mathematical background of Quatum Physics I obtained a new MSc in Mathematics and finally obtained the PhD degree in Mathematics in 2013. My research interests can be summarised in the following areas:
I work on applications of functional analysis to the description of quantum systems. This is a two way road where one can find direct applications of functional analytical tools to study and address problems in Quantum Mechanics but also a source of important and interesting mathematical problems. A corner stone of my research is the study of quadratic forms associated to differential operators. Recently we contributed to the development of the long standing open problem of the representation of non-semibounded quadratic forms. Another recent line of research is the study of evolution equations in Hilbert spaces when the generators are unbounded and when the domains of the operators depend on time. In particular I am interested in existence and stability results which connect with other of my research interests; Quantum Control.
Quantum Systems with Boundary
This is a field that studies the dynamics and kinematics of quantum systems when boundaries are present in the system. These boundaries can appear due to the finite size of the system, as physical barriers or separations between subsystems and also as a way to model impurities. This is a multidisciplinary field that combines operator theory, spectral theory, analysis on manifolds and differential geometry among others, with the theory of self-adjoint extensions of differential operators and their perturbations as the principal object of analysis.
Quantum Control on Infinite Dimensional Systems
The development of recent Quantum Computation Technologies has come with the ability to steer and control the evolution of quantum systems. The mathematical theory of quantum control has his roots in the classical geometric theory of control and has been applied with great success to finite dimensional quantum systems. However, many of the quantum systems used to perform quantum computation are infinite dimensional in nature, for instance ion traps or superconducting circuits. This has the implication that in order to apply the results of the theory of quantum control one has to consider finite dimensional approximations of these quantum systems. The study of Quantum Control Theory in infinite dimensional systems can provide the next milestone in the development of quantum technologies and has gained attention in the last decade with promising results. One of the main difficulties relies on the existence and properties of the Time Dependent Schrödinger equation, formally a linear evolution equation on a Hilbert space, when the time dependent Hamiltonians are a family of unbounded operators.