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

Viola, Joe

Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 33 (2024) no. 1, pp. 237-286

We study the relationship between the classical Hamilton flow and the quantum Schrödinger evolution where the Hamiltonian is a degree-2 complex-valued polynomial. When the flow obeys a strict positivity condition equivalent to compactness of the evolution operator, we find geometric expressions for the $L^2$ operator norm and a singular-value decomposition of the Schrödinger evolution, using the Hamilton flow. The flow also gives a geometric composition law for these operators, which correspond to a large class of integral operators with nondegenerate Gaussian kernels.

Nous étudions la rélation entre le flot hamiltonien et l’évolution quantique de Schrödinger, où l’hamiltonien est un polynôme de degré 2 à valeurs complexes. Quand le flot vérifie une hypothèse de positivité stricte (qui est équivalente à la compacité de l’opérateur d’évolution), nous trouvons des formules géométriques pour la norme de l’opérateur d’évolution agissant sur $L^2\left({ℝ}^n\right)$ ainsi qu’une décomposition en valeurs singulières de cet opérateur, en fonction du flot hamiltonien. Le flot donne aussi une loi pour la composition de ces opérateurs, qui correspondent à une grande classe d’opérateurs à noyaux gaussiens.

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