Abstract
In quantum mechanics an important role, when dealing with coherent states, is played by the so-called displacement operator, which is a suitable exponential of bosonic ladder operators. We discuss the BCH formula in connection with this operator, and with some of its generalizations.
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Notes
- 1.
For the moment, we will not make any difference between bounded and unbounded operators. This difference is, in fact, the core of the problem.
- 2.
It is enough to compute the radius of convergence of this series. This is, for each fixed k, ∞.
- 3.
Our proof is not particularly different from the one usually proposed in the literature. However, from a mathematical point of view, our version is interesting since it deals easily with domain issues.
- 4.
In other words, we prefer to call this set simply l 0 rather than, for instance, l 0(A, μ).
References
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Bagarello, F. (2022). Our Way to the BCH Formula. In: Pseudo-Bosons and Their Coherent States. Mathematical Physics Studies. Springer, Cham. https://doi.org/10.1007/978-3-030-94999-0_4
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DOI: https://doi.org/10.1007/978-3-030-94999-0_4
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