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On level crossing in deterministic and random matrix pencils

Boris Shapiro
Stockholm University
Thursday, May 3, 2018 - 16:00 to 17:30

In many spectral problems in both mathematics and physics one uses the perturbation approach which means that one has to study the spectral behavior for a pencil  $A + t B$ of linear operators, where $A$ is the initial and $B$ is the perturbing operators and t is the perturbation parameter. Of specific interest are level crossing, i.e., those values of $ t $ for which the spectrum of $A+tB$ is not simple. Level crossing usually form a very intriguing discrete set in the complex plane of parameter $ t $ which is responsible, in particular, for the convergence radius of the perturbation series expansion.In the first part of the talk I will discuss two concrete examples coming from quasi-exactly solvable models. In one of them level crossing show surprising similarities with the roots of the so-called Yablonski-Vorob’ev polynomials related to Painleve II equation. In the second part of the talk I will discuss new results about the distribution of level crossing in the situation when $A$ and $B$ are independently chosen from some standard Gaussian random matrix ensembles.

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