On quantum dissipative systems: ground states and orbital stability

Goudon, Thierry; Vivion, Léo

Journal de l’École polytechnique — Mathématiques, Tome 10 (2023), pp. 447-511

We investigate the existence and stability of ground states for a model coupling the Schrödinger equation to the wave equation in transverse directions. The model is intended to describe complex interactions between quantum particles and their environment. The result can be interpreted as a dissipation statement, induced by the energy exchanges with the environment. The proofs use either concentration-compactness arguments or spectral analysis of the linearized energy. Difficulties arise related to the fact the model does not satisfy scale invariance properties.

Nous étudions l’existence et la stabilité des états fondamentaux pour un modèle couplant l’équation de Schrödinger à l’équation d’onde dans des directions transverses. Ce modèle vise à décrire les interactions complexes entre des particules quantiques et leur environnement. Le résultat peut être interprété comme une propriété de dissipation, induite par les échanges d’énergie avec l’environnement. Les démonstrations reposent soit sur des arguments de concentration-compacité, soit sur une analyse spectrale de l’énergie linéarisée. Des difficultés surviennent liées au fait que le modèle ne satisfait pas de propriétés d’invariance d’échelle.

Classification : 35Q40, 35Q51, 35Q55
Keywords: Open quantum systems, particles interacting with a vibrational field, Schrödinger-wave equation, ground states, orbital stability
Mots-clés : Systèmes quantiques ouverts, particules en interaction avec un environnement vibratoire, système Schrödinger-ondes, états fondamentaux, stabilité orbitale

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