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.

Self-adjoint extensions and unitary operators on the boundary

Facchi, Paolo
;
Garnero, Giancarlo;Ligabò, Marilena
2018-01-01

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.
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11586/207159
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