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Squeezed states in the quaternionic setting. (English) Zbl 1436.81068

Summary: Using a left multiplication defined on a right quaternionic Hilbert space, we shall demonstrate that pure squeezed states, which are obtained by the sole action of the squeeze operator on the vacuum state, can be defined with all the desired properties on a right quaternionic Hilbert space. Further, we shall also demonstrate that squeezed states, which are obtained by the action of the squeeze operator on canonical coherent states, in other words they are obtained by the action of the displacement operator followed by the action of the squeeze operator on the vacuum state, can be defined on the same Hilbert space, but the non-commutativity of quaternions prevents us in getting the desired results. However, we will show that if one considers the quaternionic slice wise approach, then the desired properties can be obtained for quaternionic squeezed states.

MSC:

81R30 Coherent states
46E22 Hilbert spaces with reproducing kernels (= (proper) functional Hilbert spaces, including de Branges-Rovnyak and other structured spaces)
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