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arXiv · 1703.04091

Self-adjoint extensions and unitary operators on the boundary

Paolo Facchi·Giancarlo Garnero·Marilena Ligabò
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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. More- over, 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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Paolo Facchi, Giancarlo Garnero, Marilena Ligabò. 2017-03-12. Self-adjoint extensions and unitary operators on the boundary. https://doi.org/10.1007/s11005-017-1001-8

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