Abstract
We consider expressions \(\mathcal {L}_k[y]\) that are ordered products of \(k\), \(k\in \mathbb {N} \), linear symmetric quasidifferential expressions of the second order with matrix coefficients. Asymptotic formulas as \(x\to +\infty \) are derived for one of the fundamental solution systems of the equation \(\mathcal {L}_k[y]=\lambda y \), \(\lambda \in \mathbb {C} \), \(\lambda \ne 0\), under some conditions on the behavior at infinity of its coefficients. The result is applied to the study of spectral properties of operators generated by the expression \(\mathcal {L}_k[y] \), and, in particular, to second-order matrix differential operators of the Sturm–Liouville type.
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Notes
This implies that the number of Jordan blocks corresponding to the root \(z_{q+p} \) is \(s_p \) and their orders \(i_1 \), \(i_2 \), . . . , \(i_{s_p} \) satisfy the relation \(i_1+i_2+\ldots +i_{s_p}=r_p \); moreover, \(\sum _{p=1}^l r_p+q=2nk \) and \(\max \limits _{\genfrac {}{}{0pt}{1}{1\le m\le s_p}{1\le p\le l}}i_m=r+1 \).
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ACKNOWLEDGMENTS
The author is deeply indebted to K.A. Mirzoev for posing this problem and for useful discussions and to the reviewer for valuable comments and additions.
Funding
This work was supported by the joint program “Mikhail Lomonosov” of the Russian Ministry of Education and Science and DAAD under grant no. 1.12791.2018/12.2.
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Translated by V. Potapchouck
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Braeutigam, I.N. Spectral Properties of Matrix Differential Equations with Nonsmooth Coefficients. Diff Equat 56, 685–695 (2020). https://doi.org/10.1134/S0012266120060026
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DOI: https://doi.org/10.1134/S0012266120060026