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For questions about mathematical problems arising from quantum mechanics, a branch of physics describing the behaviour of nature at very small scales, at the level of atoms and subatomic particles.

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Self adjoint operator and vertex conditions in quantum graphs

A more direct proof can be found in the book "Introduction to Quantum Graphs", http://www.ams.org/bookstore-getitem/item=SURV-186, Theorem 1.4.4
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