Research Professorat the Universidad Autónoma Metropolitana Iztapalapa campus (UAM - I)

Research interests: spectral theory of linear relations (or multivalued linear operators) and its applications, quantum Markov semigroups, spectral graph theory and Jacobi operators.

Research:

The spectral theory of linear relations extends the classical theory of linear operators and, in several works, it is known as multivalued linear operators. The special issue to address this theory goes on the study of selfadjoint extensions of non-densely defined operators, which are found in so many classes of differential and difference operators, particularly, Schrödinger operators, Jacobi operators, and operators whose matrix representation is a band diagonal matrix. All these operators appear in mathematical physics and have multiple applications not only within physics.

Quantum Markov semigroups theory is studied in the sense of weak coupling limit type by using Markov generators associated with Hamiltonians. Both algebraic structure and spectral behavior of these generators, shed light on algebraic and spectral properties of the quantum semigroups generated. This study is crucial in quantum energy transport models.

The spectral graph theory is tackled due to its several applications such that transport models, quantum walks, complex network theory, quantum mechanics, among others. The graph theory is closely related to the spectral theory of linear operators in Hilbert spaces, by recognizing the set of vertices of a given graph as the canonical basis of a certain Hilbert space H. The adjacency matrix acts as a linear operator in H, and its spectrum allows analyzing the behavior of the graph.

Jacobi operators are well-studied since they can determine the classical moment problem. Besides, these operators are closely related with the theory of orthogonal polynomials since they obey a recurrence relation whose coefficients corresponds to a Jacobi sequence. A similar research line which involves these operators is the theory of interacting Fock spaces where a quantum decomposition appears, which sheds the so-call creation, annihilation and preservation operators.