Quantum controllability on graph-like manifolds through magnetic potentials and boundary conditions

Abstract

We investigate the controllability of an infinite-dimensional quantum system by modifying the boundary conditions instead of applying external fields. We analyse the existence of solutions of the Schrödinger equation for a time-dependent Hamiltonian with time-dependent domain, but constant form domain. A stability theorem for such systems, which improves known results, is proven. We study magnetic systems on Quantum Circuits, a higherdimensional generalisation of Quantum Graphs whose edges are allowed to be manifolds of arbitrary dimension. We consider a particular class of boundary conditions (quasi-δ boundary conditions) compatible with the graph structure. By applying the stability theorem, we prove that these systems are approximately controllable. Smooth control functions are allowed in this framework.

On the Schrödinger Equation for Time-Dependent Hamiltonians with a Constant Form Domain

“…Results in this direction have been found for a quantum particle in a one-dimensional cavity with moving walls, which can be transformed into a fixed-boundary problem [20][21][22][23]. More recently, time-dependent boundary conditions in a graph-like manifold, potentially relevant in quantum control theory, have been investigated [24,25].…”

“…A more recent work on the abstract non-autonomous Schrödinger problem is [34]. The stability properties of time-dependent Hamiltonians have also been investigated via non-standard analysis techniques [35] and, more recently, in [25], where sharper bounds to the norms of the solutions obtained by B. Simon were found. Finally, let us point out that many practical models use the nonlinear Schrödinger equation, i.e., they involve nonlinear terms, and a certain energy law is preserved.…”

On the Schrödinger Equation for Time-Dependent Hamiltonians with a Constant Form Domain

“…Results in this direction have been found for a quantum particle in a one-dimensional cavity with moving walls, which can be transformed in a fixed-boundary problem [19,16,7,8]. More recently, time-dependent boundary conditions in a graph-like manifold, potentially relevant in quantum control theory, have been investigated [2,1].…”

“…A more recent work about the abstract non-autonomous Schrödinger problem is [18]. The stability properties of time-dependent Hamiltonians have also been investigated via nonstandard analysis techniques [24] and, more recently, in [1], where sharper bounds to the norms of the solutions obtained by B. Simon have been found.…”