Singular surfaces associated with multiple eigenvalues and related physical effects


Singular surfaces associated with multiple eigenvalues and related physical effects

Kirillov, O.

It is common to consider that generically discrete eigenvalues of an operator are simple. However, in multiparameter operator families multiple eigenvalues are a robust phenomenon. Singular ruled surfaces were studied already by John Wallis, who in 1655 introduced his famous conical wedge known in the modern physical literature under the name of the “double coffee filter”. Later on, due to the efforts of Monge, Catalan, Plücker, Steiner and Cayley, the development of the theory of the ruled surfaces had lead to the formulation of the projective geometry. A first non-trivial physical effect related to the double semi-simple eigenvalue was discovered by Hamilton in 1833, who established that it determines a conical singularity of the dispersion surface – the Hamilton’s diabolical point (DP) – that yields a conical ray surface, which is observable in experiments with birefringent crystals as a conical refraction. In the presence of absorption and optical activity the conical singularities of the dispersion surface can transform into branch points that correspond to double eigenvalues with the Jordan block (exceptional points, EPs). This happens because the matrix determining the dispersion relation becomes a non-Hermitian one, for which an EP has a lower codimension than for a DP. In my presentation I will talk about manifestation of the multiple eigenvalues and the singular surfaces associated with them in modern physical applications such as magnetohydrodynamics dynamo and helical magnetorotational instability where the singularities determine non-trivial scaling laws and help to establish important limits for the critical parameters. I will discuss the role of the singularities in dissipation-induced instabilities on the example of the Brouwer’s problem on a heavy particle in a rotating vessel and show its connection to the modern works on crystal optics, wave propagation and rotor dynamics. Finally, I will touch the issue of the geometric phase in non-Hermitian systems.

Keywords: Instability threshold; parameters; optimization; non-smooth merit functions; multiple eigenvalues; dissipation-induced instabilities

  • Lecture (others)
    Lecture at the seminar of Prof. Roland Ketzmerick, Max-Planck-Institut für Physik komplexer Systeme (MPI-PKS), 21.01.2011, Dresden, Germany

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Publ.-Id: 16416