Abstract
We investigate locally compact topological groups for which a generalized analog of the Heisenberg uncertainty inequality hold. In particular, it is shown that this inequality holds for $\mathbb{R}^{n} \times K$ (where K is a separable unimodular locally compact group of type I), Euclidean motion group and several general classes of nilpotent Lie groups which include thread-like nilpotent Lie groups, 2-NPC nilpotent Lie groups and several low-dimensional nilpotent Lie groups.
Highlights
In, Heisenberg presented a principle related to the uncertainties in the measurements of position and momentum of microscopic particles
The Fourier transform of f ∈ L (Rn) is given by f(ξ ) = f (x)e– πi x,ξ dx, Rn where ·, · denotes the usual inner product on Rn. This definition of Fourier transform holds for functions in L (Rn) ∩ L (Rn)
Since L (Rn) ∩ L (Rn) is dense in L (Rn), the definition of Fourier transform can be extended to the functions in L (Rn)
Summary
In , Heisenberg presented a principle related to the uncertainties in the measurements of position and momentum of microscopic particles. ). , In Section , we shall prove a generalized analog of the Heisenberg uncertainty inequality for Rn × K , where K is a separable unimodular locally compact group of type I. The last section deals with a generalized analog of the Heisenberg uncertainty inequality for several general classes of nilpotent Lie groups for which the Hilbert-Schmidt norm of the group Fourier transform πξ (f ) of f attains a particular form. These classes include thread-like nilpotent Lie groups, -NPC nilpotent Lie groups and several low-dimensional nilpotent Lie groups. For any f ∈ L (Rn × K) (where K is a separable unimodular locally compact group of type I) and a, b ≥ , we have nf x a f (x, k) dk dx a
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