Mathematical Analysis and Geometry
Spectral Theory for Clifford operators:
The spectral theory based on the S-spectrum was developed to provide a rigorous foundation for quaternionic quantum mechanics, extending the work of Birkhoff and von Neumann. It has applications in various frameworks, including fractional diffusion, vector analysis, and differential geometry.
The S-spectrum is defined via a second-order polynomial of the operator in a way that
allows the holomorphic functional calculus and the spectral theorem to naturally extend to this setting. This theory also leads to generalized functional calculi for polyharmonic and polyanalytic functions, enhancing the analysis of operators and broadening the scope of operator theory beyond classical contexts.
Differential geometry and spectral theory:
This research area regards spectral theory of the Dirac operator via the S-spectrum on manifolds with particular emphasis on the hyperbolic and spherical geometries. Using this new developed spectral theory one can study bounded or unbounded operators extending in a nontrivial way the theory of Clifford operators in curved spaces.
Theory of Aharonov-Berry superoscillations:
Superoscillations are a phenomenon where a signal or wave oscillates faster than its highest frequency component, meaning it exhibits local variations at a frequency higher than the bandwidth limit of its spectrum. This occurs due to the constructive interference of lower-frequency components, creating localized zones of high-frequency behavior. Superoscillations have significant implications in areas such as Aharonov weak-values in quantum mechanics, optics, signal processing, and imaging, as they enable resolutions and manipulations beyond conventional physical and spectral constraints. The notion of supershift generalizes superoscillation, and the connection between this concept and analyticity of functions is an ongoing topic of research.
Quantum Probability
Quantum Probability is the mathematical theory that studies the laws of chance in the Quantum World.
Open Quantum Systems
Quantum Markov (or dynamical) semigroups have become the key structure for describing open system dynamics. They can also be viewed as a non-commutative generalisation of classical Markov semigroups on a commutative algebra. The study of their properties and behavior clarifies the properties of the physical systems from which the models arise. We are currently analyzing the structure and properties of quantum Markov semigroups of Gaussian type.
Quantum Information and Measurement
We investigate sequential or approximate joint measurements of complementary observables in quantum mechanics, quantum incompatibility and uncertainty relations, quantum tomography, symmetries in quantum measurement with applications of group theory and harmonic analysis.
Quantum Statistics
In many experiments and practical situations, one does not have direct access to the quantum system of interest, however, usually the system interacts with a large environment. This is initially in a certain input state and ends up being in a different output state that collects some information. We investigate estimation problems become trying of the unknown parameters of the evolution from the output state and the study of the stochastic process emerging from performing a certain measurement scheme on the output, both at the level of measurement outcomes and conditional state of the system.
Quantum dynamical systems
A general quantum dynamical system is often represented with a group of *-automorphisms on an operator *-algebra. The invariant states (the quantum analogue of the invariant measures), or the equilibrium states for the group play the central role in the understanding of the system. Quasi-invariant measures are a much wider class than invariant, and they constitute the natural environment for the theory of dynamical systems and ergodic theory. The development of a real quantum analogue of the classical theory of dynamical systems, requires developing a theory of quasi-invariant states on general operator algebras.