×

zbMATH — the first resource for mathematics

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 \(T^* \mathbb{C}^n\).

MSC:
35S05 Pseudodifferential operators as generalizations of partial differential operators
11E12 Quadratic forms over global rings and fields
35S30 Fourier integral operators applied to PDEs
PDF BibTeX XML Cite
Full Text: DOI EuDML
References:
[1] J. E. Avron and I. Herbst, Spectral and scattering theory of Schrödinger operators related to the Stark effect. Comm. Math. Phys. 52 (1977) 239–254 · Zbl 0351.47007
[2] R. Cushman and J. J. Duistermaat, The behavior of the index of a periodic linear Hamiltonian system under iteration. Advances in Math. 23 1–21 (1977) · Zbl 0362.58005
[3] Jan Derezinski, Some remarks on Weyl pseudodifferential operators. Journées Équations aux dérivées partielles Saint-Jean-de-Monts (1973) XII:1–14
[4] Lars Hörmander, The analysis of linear partial differential operators III. Springer Verlag 1985 · Zbl 0601.35001
[5] Lars Hörmander,L 2 estimates for Fourier integral operators with complex phase. Arkiv för matematik 21 (1983), 283–307 · Zbl 0533.47045
[6] John Williamson, On the algebraic problem concerning the normal forms of linear dynamical systems. Amer. J. Math. 58 (1936), 141–163 · Zbl 0013.28401
[7] K. Yosida, Functional analysis. Springer Verlag 1968, Grundl. d. math. Wiss. 123 · Zbl 0152.32102
[8] A. J. Laub and K. Meyer, Canonical forms for symplectic and Hamiltonian matrices. Celestial Mech. 9 (1974), 213–238 · Zbl 0316.15005
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.