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We study the relationship between the classical Hamilton flow and the quantumSchr\"odinger evolution where the Hamiltonian is a degree-2 complex-valuedpolynomial. When the flow obeys a strict positivity condition equivalent tocompactness of the evolution operator, we find geometric expressions for the$L^2$ operator norm and a singular-value decomposition of the Schr\"odingerevolution, using the Hamilton flow. The flow also gives a geometric compositionlaw for these operators, which correspond to a large class of integraloperators with nondegenerate Gaussian kernels.

The elliptic evolution of non-self-adjoint degree-2 Hamiltonians