For a periodic quantum circuit made of a chain with a loop at each node, translation-invariant boundary conditions force the coupling parameter δ and the loop phase difference α to be constant along the whole chain.
Boundary dynamics and topology change in quantum mechanics
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abstract
We show how to use boundary conditions to drive the evolution on a Quantum Mechanical system. We will see how this problem can be expressed in terms of a time-dependent Schr\"{o}dinger equation. In particular we will need the theory of self-adjoint extensions of differential operators in manifolds with boundary. An introduction of the latter as well as meaningful examples will be given. It is known that different boundary conditions can be used to describe different topologies of the associated quantum systems. We will use the previous results to study how this topology change can be accomplished in a dynamical way.
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On $\mathbb{Z}$-invariant self-adjoint extensions of the Laplacian on quantum circuits
For a periodic quantum circuit made of a chain with a loop at each node, translation-invariant boundary conditions force the coupling parameter δ and the loop phase difference α to be constant along the whole chain.