Essential Spectrum Due to Singularity

The aim of this short note is to explain important spectral phenomenon which can be observed for differential operators appearing in problems of magnetic hydrodynamics. This phenomenon are due to the nonperturbative behavior of the essential spectrum. It has been observed that the essential spectrum of a matrix differential operator on a finite interval can be different from the limit of the essential spectra of the differential operators determined by the same operator matrix on an increasing sequence of smaller intervals tending to the original interval. Trivial counterpart of this phenomena is well known for infinite intervals, since for example the essential spectrum of − d2 dx2 on a finite interval [−an, bn] is empty and therefore does not give the essential spectrum of − d 2 dx2 on the whole line when an, bn → ∞. The phenomenon described in the current note is more sophisticated and is due to a rather complicated interplay between the components of