Asymptotics of solutions of matrix differential equations with nonsmooth coefficients. (English. Russian original) Zbl 1404.34095
Math. Notes 104, No. 1, 150-155 (2018); translation from Mat. Zametki 104, No. 1, 148-153 (2018).
From the text: In this paper, we consider matrix second-order differential expressions \(l[y]\) with generally nonsmooth coefficients, formal products \({\mathcal L}_k[y]\) of such expressions, and the operators generated by them on the space \(L^2_n({\mathcal I})\), where \({\mathcal I}:= [a, +\infty)\) \((a>0)\).
Under certain constraints on the coefficients of these expressions (see Theorem 1), we obtain asymptotic formulas for a fundamental system of solutions of the equation \({\mathcal L}_k[y]= \lambda_y\) for fixed \(\lambda\) as \(x\to+\infty\). The results obtained are used to find the deficiency numbers of the corresponding operators and study the spectra of their self-adjoint extensions (see Theorems 2 and 3).
Under certain constraints on the coefficients of these expressions (see Theorem 1), we obtain asymptotic formulas for a fundamental system of solutions of the equation \({\mathcal L}_k[y]= \lambda_y\) for fixed \(\lambda\) as \(x\to+\infty\). The results obtained are used to find the deficiency numbers of the corresponding operators and study the spectra of their self-adjoint extensions (see Theorems 2 and 3).
MSC:
| 34L05 | General spectral theory of ordinary differential operators |
| 34L15 | Eigenvalues, estimation of eigenvalues, upper and lower bounds of ordinary differential operators |
| 34D05 | Asymptotic properties of solutions to ordinary differential equations |
| 34A05 | Explicit solutions, first integrals of ordinary differential equations |
| 34B09 | Boundary eigenvalue problems for ordinary differential equations |
| 47E05 | General theory of ordinary differential operators |
Keywords:
quasiderivative; quasidifferential expression; asymptotics of the fundamental system of solutions; matrix differential operators; deficiency numbers; spectrumReferences:
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