Boris Shapiro: Level crossings in deterministic and random matrix pencils

Abstract: Collisions of energy levels (alias <q>level crossings</q>) in pencils of linear operators is a physically important phenomenon studied since the down of quantum mechanics. In this talk we discuss two intriguing concrete examples coming from the so-called quasi-exactly solvable potentials and mainly concentrate on some recent results about the distribution of level crossings in families A+t B, where A and B and independently taken from some standard random matrix ensembles.