Self-adjoint extensions and unitary operators on the boundary

Abstract

Abstract. We establish a bijection between the self-adjoint extensions of the Laplace operator on a bounded regular domain and the unitary operators on the boundary. Each unitary encodes a specific relation between the boundary value of the function and its normal derivative. This bijection sets up a characterization of all physically admissible dynamics of a nonrelativistic quantum particle confined in a cavity. Moreover, this correspondence is discussed also at the level of quadratic forms. Finally, the connection between this parametrization of the extensions and the classical one, in terms of boundary self-adjoint operators on closed subspaces, is shown.

Quantum cavities with alternating boundary conditions

“…We are now ready to state the following result, proved in [7], about the self-adjoint extensions of T .…”

“…It was proved in [6,7], that every unitary operator U acting on H b uniquely determines a well-defined physical realization of T , i.e a self-adjoint extension of T , denoted by T U , identifying some boundary conditions. The relation between the selfadjoint extension T U and the unitary operator U will be recalled in Sections 3 and 4.…”

“…Once these aspects are clarified, the one-to-one correspondence between the unitary operators acting on H b and the self-adjoint extensions of the operator T will hold as well as in the one-dimensional case. See [7].…”

“…Since we are going to describe a free particle in a cavity Ω we need some results about self-adjoint extensions of the operator T in terms of boundary conditions. This was largely discussed in [3,6,7].…”

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.…”