# Self-adjoint extensions and unitary operators on the boundary

## TL;DR

This paper establishes a bijection between self-adjoint extensions of the Laplace operator on a domain and boundary unitary operators, providing a comprehensive framework for quantum boundary conditions and dynamics.

## Contribution

It introduces a novel correspondence linking boundary unitary operators with self-adjoint extensions, enhancing understanding of quantum boundary conditions.

## Key findings

- Bijection between self-adjoint extensions and boundary unitaries
- Characterization of quantum dynamics via boundary relations
- Connection to classical boundary operator parametrizations

## 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.