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 quasiexactly solvable models. In one of them level crossing show surprising similarities with the roots of the socalled YablonskiVorob’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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On level crossing in deterministic and random matrix pencils
Research Group:
Boris Shapiro
Institution:
Stockholm University
Location:
A136
Schedule:
Thursday, May 3, 2018  16:00 to 17:30
Abstract:
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