Boundary dynamics and topology change in quantum mechanics

Abstract

We show how to use boundary conditions to drive the evolution on a Quantum Mechanical system. We will see how this problem can be expressed in terms of a time-dependent Schr\"{o}dinger equation. In particular we will need the theory of self-adjoint extensions of differential operators in manifolds with boundary. An introduction of the latter as well as meaningful examples will be given. It is known that different boundary conditions can be used to describe different topologies of the associated quantum systems. We will use the previous results to study how this topology change can be accomplished in a dynamical way

Quantum magnetic billiards: boundary conditions and gauge transformations

“…The study of quantum systems confined in a bounded domain requires a careful description of the physical properties of its boundary, and thus of the interaction between the system and the boundary, effectively encoded via a proper choice of boundary conditions. In recent years, quantum boundary conditions have increasingly attracted interest in different branches of quantum physics [1,2], some examples being the analysis of quantum Hall systems [3,4], the study of geometric phases [5], quantum control theory, topological insulators and QCD [6], quantum gravity and topology change [7,8], as well as the Casimir effect in quantum field theory [9,10,11], to name a few.…”

Field Fluctuations and Casimir Energy of 1D-Fermions

“…In particular, those works have applied the theory of self-adjoint extensions to the computation of the so-called Casimir energy of scalar fields [8,9]. On the other hand, Ibort et al have developed the theory of local self-adjoint extensions for Dirac-type operators, and have studied applications in condensed matter [10][11][12]. More recent works have shown how the theory of self-adjoint extensions can be used to describe physical systems confined to cavities [13,14].…”

Quantum Control at the Boundary

“…The QCB paradigm has been used to show how to generate entangled states in composite systems by suitable modifications of the boundary conditions [30]. The relation of QCB and topology change has been explored in [39] and recently used to describe the physical properties of systems with moving walls ( [18], [19], [16], [17], [21]), but in spite of its intrinsic interest some basic issues such as the QCB controllability of simple systems has never been addressed.…”

“…In developing the theory it will be shown first, by means of a suitable chosen timedependent unitary transformation, that the variation of the boundary conditions of the system can be implemented as a time-dependent family of Hamiltonian operators, an idea that was already anticipated in [39]. The particular instance of quasi-periodic boundary conditions will be worked out explicitly and it will be shown that the system reduces to a linear system similar to those studied by Chambrion et al [11].…”