Proof of the Lieb-Thirring conjecture in the case of strictly convex quadratic potential. (Preuve de la conjecture de Lieb-Thirring dans le cas des potentiels quadratiques strictement convexes.) (French) Zbl 0938.35148

de la Bretèche, R.

Ann. Inst. Henri Poincaré, Phys. Théor. 70, No. 4, 369-380 (1999).

The author proves the Lieb-Thirring conjecture in the particular case of positive definite quadratic potentials. More precisely, if \(V(x)=\sum k_i^2x_i^2\) with \(k_i>0\), it is shown that for any convex function \(\varphi\) on \({\mathbb R}\) and \(E\in {\mathbb R}\) one has: \[ \sum_{m\in{\mathbb N}^n}\varphi (E-\sum_i (2m_i+1)k_i)\leq (2\pi)^{-n}\int \varphi (E-\xi^2 -V(x)) dx d\xi . \] The proof is based on a reduction to the one-dimensional case and elementary properties of convex functions.

Reviewer: André Martinez (Bologna)

MSC:

35Q40	PDEs in connection with quantum mechanics
35P20	Asymptotic distributions of eigenvalues in context of PDEs
81Q05	Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics

Keywords:

harmonic oscillator; Schrödinger operator; convex functions

References:

[1]	M. Aizenman et E. Lieb , On semi-classical bounds for eigenvalues of Schrödinger Operators , Phys. Lett. 66A , 1978 , 427 - 429 . MR 598768
[2]	Ph Blanchard et J. Stubbe , Bound States for Schrödinger Hamiltonians : Phase space methods and applications , Rev. in Math. Physics , vol. 8 , n^\circ 4 , 1996 , 503 - 548 . MR 1405763 | Zbl 0859.35101 · Zbl 0859.35101
[3]	B. Helffer et B. Parisse , Riesz means of bound states and semi classical limit connected with a Lieb-Thirring’s conjecture III , Prépublication de l’École Normale supérieure , 1990 . · Zbl 0717.35062
[4]	B. Helffer et D. Robert , Calcul fonctionnel par la transformée de Mellin et applications , J. Funct. Anal. , 53 , 1983 , 246 - 268 . MR 724029 | Zbl 0524.35103 · Zbl 0524.35103
[5]	B. Helffer et D. Robert , Riesz means of bound states and semi classical limit connected with a Lieb-Thirring’s conjecture I , Asymptotic Analysis , 3 , 91 - 103 , 1990 . MR 1061661 | Zbl 0717.35062 · Zbl 0717.35062
[6]	B. Helffer et D. Robert , Riesz means of bound states and semi classical limit connected with a Lieb-Thirring’s conjecture II , Ann. Inst. Henri Poincaré, section Physique Théorique , 53 ( 2 ), 1990 , 139 - 147 . Numdam | MR 1079775 | Zbl 0728.35078 · Zbl 0728.35078
[7]	A. Laptev , Dirichlet and Neumann eigenvalue problems on domains in euclidean spaces , J. Funct. Anal. , 151 , 1997 . MR 1491551 | Zbl 0892.35115 · Zbl 0892.35115
[8]	A. Laptev , On the Lieb-Thirring conjecture for a class of potentials , preprint 1997 . MR 1747896 · Zbl 0938.35112
[9]	E.H. Lieb et W.E. Thirring , Inequalities for the moments of the eigenvalues of the Schrödinger equation and their relation to Sobolev inequalities , Studies in Math. Phys. , (E. Lieb, B. Simon, A. Wightman Eds), Princeton Univ. Press , 1976 , 269 - 303 . Zbl 0342.35044 · Zbl 0342.35044
[10]	H. Matsumoto et N. Ueki , Spectral Analysis of Schrödinger operators with Magnetic Fields , J. Funct. Anal. , 140 , n^\circ 1 , 218 - 255 . MR 1404582 | Zbl 0866.35083 · Zbl 0866.35083