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Symplectic classification of quadratic forms, and general Mehler formulas. (English) Zbl 0829.35150
Summary: The first part of the paper gives a complete classification of real (complex) quadratic forms in a real (complex) symplectic vector space, with explicitly listed normal forms. For complex valued quadratic forms \(Q\) in \(T^* \mathbb{R}^n\) such that \(\text{Re} Q \leq 0\) in \(T^* \mathbb{R}^n\), the Weyl symbol of the semigroup \(\exp (tQ^w (x,D))\), \(t \geq 0\), generated by the corresponding Weyl operator \(Q^w (x,D)\) is then determined; it is always a Gaussian multiplied by a Lebesgue measure in the range of \(\cos tF\) where \(F\) is the Hamilton map corresponding to \(Q\). The classical Mehler formula corresponds to the harmonic oscillator. The determination of the sign of the symbol relies on the classification in the first part. An extension to inhomogeneous quadratic forms contains the Avron-Herbst formula for the Stark operator.
Finally, the results are interpreted in terms of the calculus of generalized Gaussians regarded as infinitesimal Fourier integral operators, and the semigroup generated by the operators \(\exp (Q^w(x,D))\) with \(\text{Re} Q \leq 0\) is identified with the operators with symbol \(\pm 1\) corresponding to positive symplectic maps in \(T^* \mathbb{C}^n\).

35S05 Pseudodifferential operators as generalizations of partial differential operators
11E12 Quadratic forms over global rings and fields
35S30 Fourier integral operators applied to PDEs
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