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Gaussian kernels representing operators on the Hilbert space scrH=L^2 (openR^n) are studied. Necessary and sufficient conditions on such a kernel in order that the corresponding operator be positive semidefinite, corresponding to a density matrix (cross-spectral density) in quantum mechanics (optics), are derived. The Wigner distribution method is shown to be a convenient framework for characterizing Gaussian kernels and their unitary evolution under Sp (2n, openR) action. The nontrivial role played by a phase term in the kernel is brought out. The entire analysis is presented in a form which is directly applicable to n-dimensional oscillator systems in quantum mechanics and to Gaussian Schell-model partially coherent fields in optics.
Simon et al. (Thu,) studied this question.
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