Critical layer singularities and complex eigenvalues in some differential equations of mathematical physics

Details Critical layer singularities and complex eigenvalues in some differential equations of mathematical physics Critical layer singularities and complex eigenvalues in some differential equations of mathematical physics

Sep 1986

adsabs.harvard.edu/cgi-bin/nph-data_query?bibcode=1986phr...142..263a&link_type=abstract

Physics Reports, Volume 142, Issue 5, p. 263-356.

Physics

23

Scientific paper

Singular differential equations are a common feature of many problems in mathematical physics. It is often the case that systems with a similar mathematical structure can arise in many different contexts. In this article, mathematically related problems are drawn from areas as diverse as hydrodynamics (with applications to oceanography and meteorology), magnetohydrodynamics and plasma physics (with applications to astrophysics and geophysics, especially solar physics, ionospheric and magnetospheric physics; also nuclear fusion devices), acoustics, electromagnetics, quantum mechanics and nuclear physics. One major unifying feature common to the problems discussed here is the existence of complex eigenvalues, often associated with so-called ``classical self-adjoint'' equations. No real contradiction is involved here, but the resulting wave functions are often referred to variously as ``radioactive states'', ``damped resonances'', ``leaky waves'', ``non-modal solutions'' , ``singular modes'', ``virtual modes'', or ``improper eigenfunctions''. In the hydrodynamics of shear flows, such modes are associated with the existence of ``critical layers'' at which a singularity occurs in the governing (ordinary) differential equation. Similar, but usually more general singular layers are known to occur in equations arising in many of the above-mentioned contexts, and it is the purpose of this review to identify the nature of these singular layers and complex eigenvalues, and the relationships that exist between the different context in which they are found, and in particular to emphasize the occurrence of and interpretation of complex eigenvalues in quantum mechanics. Thus the ``exponential catastrophe'' is a clearly identified and recurring theme throughout this article by virtue of the similarities that exist between the classical and quantum system discussed here. The examples quoted from quantum mechanics are simple in form, and found in many standard texts, but the virtue of including them here is twofold: the results are easy to understand and relate to the more complicated ``classical'' systems, and they provide a valuable didactic and pedagogic tool for those readers whose background in quantum mechanics is limited. It is also hoped that this article will be of interest to readers who wish to become more acquainted with some aspects of hydrodynamics and magnetohydrodynamics.

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