Limit theorems for the maximal eigenvalues of the mean-field Hamiltonian with random potential

Let , be the mean-field Hamiltonian with and random i.i.d. potential ξ_V. We prove limit theorems for the extreme eigenvalues of as |V|→∞. The limiting distributions are the same as for the corresponding extremes of ξ_V only if either (i) ξ_V is undbounded and , or (ii) ξ_V is bounded with “sharp” peaks and . Localization properties for the corresponding eigenfunctions are also studied. Institute of Mathematics and Informatics, Akademijos 4, 2600 Vilnius, Lithuania. Published in Lietuvos Matematikos Rinkinys, Vol. 39, No. 2, pp. 147–168, April–June, 1999.