Exceptional points, spectral singularities, vector field singularities and discriminant varieties


Exceptional points, spectral singularities, vector field singularities and discriminant varieties

Günther, U.

Many physically realistic problems can be described in terms of parameter-dependent eigenvalue problems of some non-Hermitian operators, as scattering problems or as dynamical flow problems. A common feature of such problems is the existence of degenerate configurations which can be associated to dynamical bifurcation behavior, spectral branch points or special types of singularities in the scattering matrix. Mathematically, such problems are related to non-diagonalizable spectral operator decompositions (the existence of non-trivial Jordan blocks), multiple eigenvalues and the coalescence of singularities in vector flow fields. In parameter spaces these configurations show up as so called discriminant varieties well known from algebraic geometry and singularity theory. In the talk, the structural interrelation of these effects is demonstrated and illustrated on concrete problems from PT quantum mechanics, optical lasing systems and the Bloch-sphere representation of simple time-dependent quantum mechanical problems.

Keywords: non-Hermitian operators; degenerate configurations; dynamical bifurcation behavior; spectral branch points; spectral singularities; Jordan blocks; multiple eigenvalues; singularity coalescence; discriminant varieties

  • Invited lecture (Conferences)
    Mathematics in Technical and Natural Sciences, 18.-24.09.2015, Koscielisko, Poland

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Publ.-Id: 22456