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The elliptic evolution of non-self-adjoint degree-2 Hamiltonians
Joe Viola1
1 Nantes Université, Laboratoire de Mathématiques Jean Leray, LMJL, F-44000 Nantes, France
Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 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 (ℝ n ) 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.

Received:
Accepted:
Published online:
DOI: 10.5802/afst.1770

Joe Viola 1

1 Nantes Université, Laboratoire de Mathématiques Jean Leray, LMJL, F-44000 Nantes, France
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights 
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Joe Viola. The elliptic evolution of non-self-adjoint degree-2 Hamiltonians. Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 33 (2024) no. 1, pp. 237-286. doi : 10.5802/afst.1770. https://afst.centre-mersenne.org/articles/10.5802/afst.1770/

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