FoQuS

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.

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.

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 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 study quantum Markov semigroups from different perspectives. On the one hand, we are currently analyzing the structure and properties of quantum Markov semigroups of Gaussian type. On the other hand, we are studying quantum Markov semigroups describing the thermalization dynamcis of quantum spin systems, especially in the context of topological phases of matter in 2D.

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.

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.

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.

The analysis of the spectral properties of atoms and crystalline structures lies at the core of the understanding of the electrochemical features of solids and represents the earliest successful applications of quantum mechanics. The discovery of the quantum Hall effect in 1980 paved the way to the study of topological phases of matter, among which topological insulators are the prominent example. A comprehensive description of such exotic materials requires an interplay of spectral theory, geometry and topology. Motivated by this framework, we investigate the spectral and transport properties of magnetic Schrödinger operators, as well as their tight-binding counterparts, in the one-particle approximation. In particular, we are interested in the analysis of bulk-edge correspondence and in the study of topological properties via the study of Wannier-like functions. Furthermore, we investigate the stability properties of the Laughlin phase describing the fractional quantum Hall effect.

Particles obeying to exotic fractional statistics (anyons) are known to be forbidden by the postulates of quantum mechanics in the usual three-dimensional world. However, as observed in fractional quantum Hall physics, two-dimensional quasi-particles, i.e., excitations of certain low-dimensional quantum systems, may carry a fractional charge and behave as anyons.

In a suitable representation (magnetic gauge), anyonic particles are described as conventional bosons carrying a singular magnetic flux of Aharonov-Bohm (AB) type in terms. The investigation of the well-posedness of such magnetic Schrödinger operators (as suitable self-adjoint realizations) as well as their spectral and scattering properties is highly non-trivial, even in the non-interacting case. We plan to address such questions in different physical settings.

Bell-type inequalities poses the issue whether, and how, locality and separability hold in the quantum mechanics. In quantum field theory, correlations between space-like separated entangled systems even appear in conflict with Einstein’s condition of relativist causality, whereby no causal process can propagate faster than light. We distinguish and classify different formalizations of non-locality and causality and we show in what sense such notions are compatible within the theory.

Macroscopic phenomena are observed to take place just in one direction of time, that is from past to future, in accordance with thermodynamical irreversibility. Nevertheless, the underlying microscopic dynamics is time-reversal invariant, which raises the philosophical problem whether the arrow of time can be grounded at the quantum level. We explore and develop two alternative proposals to recover irreversibility: one aims to add some time-asymmetrical microscopic condition, which must be physically justified; whereas the other one attempts to modify the mathematical form of the microscopic dynamical evolutions, so as to make it genuinely irreversible.

The limited experimental access below the Planck scale obscures our theoretical understanding of atomic and sub-atomic matter. For one, the infamous measurement problem threatens realistic interpretations of quantum states already in non-relativistic quantum mechanics. Yet, the issue becomes even more puzzling in the relativistic regime, since it is far from settled whether the fundamental entities in quantum field theory are particles, fields or perhaps even more exotic objects. We evaluate arguments in favor and against different interpretations of the quantum world based on mathematical theorems, with the aim to elaborate a coherent ontology for the quantum world.