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Binding of Schrödinger particles through conspiracy of potential wells

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Binding of Schrödinger particles through conspiracy of potential wells ANNALES DE L’I. H. P., SECTION A M. KLAUS B. SIMON Binding of Schrödinger particles through conspiracy of potential wells Annales de l’I. H. P., section A, tome 30, no 2 (1979), p. 83-87. <http://www.numdam.org/item?id=AIHPA_1979__30_2_83_0> © Gauthier-Villars, 1979, tous droits réservés. L’accès aux archives de la revue « Annales de l’I. H. P., section A », implique l’accord avec les conditions générales d’utilisation (http://www. numdam.org/legal.php). Toute utilisation commerciale ou impression systé- matique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/ 83 Binding of Schrödinger Particles Through Conspiracy of Potential Wells M. KLAUS (*) and B. SIMON (**) Department of Physics, Princeton University, Princeton, NJ 08540 Ann. Inst. Henri Poincaré Vol. XXX, n° 2, 1979, Section A : Physique , théorique. ABSTRACT. - We study the ground state energy E(R) for when V and W are négative with compact support. In particular, in dimension 3, when - a + V and - A + W have no bound states but both have zéro energy résonances, we prove that E(R) ~ 2014 for R large with ~3 = . .321651512... In this note we want to discuss some properties of the ground state energy, E(R), of the Schrödinger operator on where V and W have compact support and lie in == ~ for v ~ 3, ~ 2 so that V(x) and W(R - x) have disjoint supports. Our first result is (all proofs deferred until later) : THEOREM 1. - Let V, W be négative. In the région R &#x3E; Ro, 1 E(B) 1 decreases as R increases, i. e. (*) Supported by Swiss National Science Foundation ; on leave from the Universityof Zurich. ’ (**) Research partially supported by USNSF Grant MCS-78-01885 also at Dept.of Mathematics. ’ Annales de l’Institut Henri Poin

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