Exponential convergence of the first eigenvalue divided by the dimension, for certain sequences of Schrödinger operator. (English) Zbl 0796.35123

Sjöstrand, Johannes

Robert, D. (ed.), Méthodes semi-classiques, Volume 2. Colloque international (Nantes, juin 1991). Paris: Société Mathématique de France, Astérisque. 210, 303-326 (1992).

This article is devoted to the study of the Schrödinger operator: \[ - h^ 2 \Delta + V^{(m)} (x) \text{ on } \mathbb{R}^ m, \] where \(V^{(m)}\) is a smooth real-valued convex potential with a structure related to the statistical mechanics (for example \(V^{(m)} (x) = \sum^ m_{k=1} v(x_ k) + I(x_ k,x_{k+1})\) with the convention that \(x_{m+1} = x_ 1\) and where \(v\) has a quadratic behavior at \(\infty\) and \(I\) defines an “interaction” between nearest neighbors controlled by \(v)\). Under suitable rather weak conditions, it is possible to prove that the first eigenvalue of the Schrödinger operator divided by the dimension tends to a limit as the dimension tends to \(\infty\).
The author gives rather natural assumptions under which the convergence appears to be exponentially rapid and obtains a measure of the rate of the exponential decay which seems to be optimal. The proof is based on exponentially weighted estimates of certain Hessians of the logarithm of the first eigenfunction and uses systematically the maximum principle in a way which was initially inspired by the article of I. M. Singer, B. Wong, S.-T. Yau and S. S.-T. Yau [Ann. Sc. Norm. Super. Pisa, Cl. Sci., IV. Ser. 12, 319-333 (1985; Zbl 0603.35070)]. Connected results but in the semiclassical context were analyzed by Helffer and Sjöstrand in the same volume.
For the entire collection see [Zbl 0778.00035].

Reviewer: B.Helffer (Paris)

MSC:

35P15	Estimates of eigenvalues in context of PDEs
35Q40	PDEs in connection with quantum mechanics
82B21	Continuum models (systems of particles, etc.) arising in equilibrium statistical mechanics

Keywords:

convergence; rate of the exponential decay; first eigenfunction; maximum principle

Citations:

Zbl 0603.35070