Abstract:
In this seminar I will present some of my research activities related to the rigorous analysis of a quantum particle confined in a bounded region of space. In the first part of the seminar I will focus on a free quantum particle, living in the one-dimensional segment, considering all the possible self-adjoint extensions of the Hamiltonian operator. The latter are described in terms of suitable boundary conditions. In particular, I will discuss both the emergence of a non-trivial unitary equivalence between different self-adjoint extensions, that is connected with the problem of "hearing" a given extension, and how some quantum boundary conditions persist when we perform the classical limit in the phase-space. In the second part of the seminar I will then move to a simple model of the quantum Hall effect, described by the magnetic Laplacian acting in an infinitely long two-dimensional strip. For this system I will describe how the quantization of the Hall conductivity is affected by finite-size effects and by non-conventional boundary conditions, and also how the latter are associated with a new kind of quantum states having no classical counterpart.

[Il seminario si svolgerà all'interno delle attività del progetto PRIN 2022AKRC5P "Interacting Quantum Systems: Topological Phenomena and Effective Theories" finanziato dall’Unione europea – Next Generation EU.]

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