LE3 .A278 2015

2015

Teismann, Holger

Acadia University

Bachelor of Science

Honours

Mathematics and Statistics

Mathematics & Statistics

Controllability, in reference to the Schr odinger equation, refers to the process of taking an initial quantum state and transforming it to a desired target state in a nite passage of time. This paper focuses on in nite-dimensional exact controllability of the linearized Schr odinger equation using a speci c result of the in nite-dimensional theory. Although there are a number of assumptions which must be satis ed for the application of the theorem to a given quantum potential, two assumptions in particular are examined here: the gap condition for the eigenvalues of the Hamiltonian and the decay condition for the sequence of matrix elements 􀀀k = h 0; ki, where (x) is the control function and k(x) is the kth eigenfunction of the Hamiltonian. These two assumptions are readily checked for a variety of potentials in order to gain an intuition about which classes of potentials might satisfy the criteria of the controllability theorem. By testing a variety of potentials it was found that the gap condition is easily satis ed for the majority of potentials whereas the decay condition is much harder to satisfy. It is concluded that the practicality of the theorem may be limited due to the number of potentials satisfying the decay condition, but that potentials which have a close geometric proximity to the in nite square well potential may have a better chance of satisfying the decay condition. The observations of this investigation allows for the possibility of nding new quantum potentials which are exactly controllable.

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https://scholar.acadiau.ca/islandora/object/theses:1260