The Cauchy problem for the Schrödinger equation in dimension three with concentrated nonlinearity

  • Gianfausto Dell'Antonio

    Dipartimento di Matematica, Università di Roma “La Sapienza”, Italy, Laboratorio Interdisciplinare SISSA-ISAS, Trieste, Italy
  • Rodolfo Figari

    Dipartimento di Scienze Fisiche, Università di Napoli “Federico II”, Italy
  • Alessandro Teta

    Dipartimento di Matematica Pura e Applicata, Università di L’Aquila, Italy
  • Riccardo Adami

    Département de mathématiques et applications, École Normale Supérieure, Paris, France

Abstract

We consider the Schrödinger equation in with nonlinearity concentrated in a finite set of points. We formulate the problem in the space of finite energy V, which is strictly larger than the standard H1-space due to the specific singularity exhibited by the solutions. We prove local existence and, for a repulsive or weakly attractive nonlinearity, also global existence of the solutions.

Résumé

On considère l’équation de Schrödinger avec une nonlinéarité concentrée en un nombre fini de points. On formule le problème dans l’espace V d’énergie finie, qui contient strictement l’espace standard H1, à cause de la singularité spécifique des solutions. On prouve des résultats d’existence locale et même, pour une nonlinéarité répulsive ou faiblement attractive, l’existence globale des solutions.

Cite this article

Gianfausto Dell'Antonio, Rodolfo Figari, Alessandro Teta, Riccardo Adami, The Cauchy problem for the Schrödinger equation in dimension three with concentrated nonlinearity. Ann. Inst. H. Poincaré Anal. Non Linéaire 20 (2003), no. 3, pp. 477–500

DOI 10.1016/S0294-1449(02)00022-7