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 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.
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 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.
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Joe Viola 1
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@article{AFST_2024_6_33_1_237_0, author = {Joe Viola}, title = {The elliptic evolution of non-self-adjoint degree-2 {Hamiltonians}}, journal = {Annales de la Facult\'e des sciences de Toulouse : Math\'ematiques}, pages = {237--286}, publisher = {Universit\'e Paul Sabatier, Toulouse}, volume = {Ser. 6, 33}, number = {1}, year = {2024}, doi = {10.5802/afst.1770}, language = {en}, url = {https://afst.centre-mersenne.org/articles/10.5802/afst.1770/} }
TY - JOUR AU - Joe Viola TI - The elliptic evolution of non-self-adjoint degree-2 Hamiltonians JO - Annales de la Faculté des sciences de Toulouse : Mathématiques PY - 2024 SP - 237 EP - 286 VL - 33 IS - 1 PB - Université Paul Sabatier, Toulouse UR - https://afst.centre-mersenne.org/articles/10.5802/afst.1770/ DO - 10.5802/afst.1770 LA - en ID - AFST_2024_6_33_1_237_0 ER -
%0 Journal Article %A Joe Viola %T The elliptic evolution of non-self-adjoint degree-2 Hamiltonians %J Annales de la Faculté des sciences de Toulouse : Mathématiques %D 2024 %P 237-286 %V 33 %N 1 %I Université Paul Sabatier, Toulouse %U https://afst.centre-mersenne.org/articles/10.5802/afst.1770/ %R 10.5802/afst.1770 %G en %F AFST_2024_6_33_1_237_0
Joe Viola. The elliptic evolution of non-self-adjoint degree-2 Hamiltonians. Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 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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