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On the Essential Spectrum of a Class of Singular Matrix Differential Operators. I: Quasiregularity Conditions and Essential Self-adjointness

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Abstract

The essential spectrum of singular matrix differential operator determined by the operator matrix

$$\left( \begin{gathered} - \frac{{d}}{{{d}x}}p(x)\frac{{d}}{{{d}x}} + q(x){ }\frac{{d}}{{{d}x}}\frac{\beta }{x} \hfill \\ { - }\frac{\beta }{x}\frac{{d}}{{{d}x}}{ }\frac{{m(x)}}{{x^2 }} \hfill \\ \end{gathered} \right)$$

is studied. It is proven that the essential spectrum of any self-adjoint operator associated with this expression consists of two branches. One of these branches (called regularity spectrum) can be obtained by approximating the operator by regular operators (with coefficients which are bounded near the origin), the second branch (called singularity spectrum) appears due to singularity of the coefficients.

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Authors
  1. Pavel Kurasov
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  2. Serguei Naboko
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Kurasov, P., Naboko, S. On the Essential Spectrum of a Class of Singular Matrix Differential Operators. I: Quasiregularity Conditions and Essential Self-adjointness. Mathematical Physics, Analysis and Geometry 5, 243–286 (2002). https://doi.org/10.1023/A:1020929007538

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