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).
Reviewer: G.Radhakrishnan


44A15 Special integral transforms (Legendre, Hilbert, etc.)
81-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to quantum theory
46F12 Integral transforms in distribution spaces
81Q10 Selfadjoint operator theory in quantum theory, including spectral analysis
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