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Continuity statements and counterintuitive examples in connection with Weyl quantization. (English) Zbl 0541.44001
The Weyl transform of a function f(p,q) on the classical phase space defines an operator on the quantum mechanical Hilbert space via: $Qf=2^{-\nu}\int_{E}dv\tilde f(v)W(-v/2),$ where $$\tilde f(v)=2^{- \nu}\int dv'e^{i\sigma(v,v')}f(v')$$, $$W(p,q)=\exp [i(p\cdot Q-q\cdot P)]$$, $$\sigma((q,p),(q',p'))={1\over2}(p\cdot q'-q\cdot p')$$ and $$(p,q)=v\in E$$, the 2$$\nu$$-dimensional phase space and $$P_ j,Q_ j$$ are the momentum and position operators respectively. This paper is concerned with the continuity properties of the Weyl correspondence. In proving their results the author uses the properties of an integral transform connecting the function with the matrix elements of Qf between coherent states: $Qf=\int_{E}da\int_{E}db| \Omega^ a)\int_{E}dvf(v)2^{\nu}(\Omega^ a,W(2v)\pi \Omega^ b)(\Omega^ b|,$ where $$\forall a\in E:\Omega^ a=W(a)\Omega,$$ and $$\Omega$$ is the ground state of the harmonic oscillator obeying $$(P_ j-iQ_ j)\Omega =0,\forall_ j$$ and $$\pi$$ is the parity operator.
After proving several continuity results the author exhibits some complementary results about what continuity properties cannot be expected. For example she shows that the Weyl transform of a positive function can be non-positive and similarly the Weyl transform of positive bounded function may be unbounded (even below).