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Discreteness of the spectrum of a polar differential operation in the space of vector functions. (English. Russian original) Zbl 1198.34033
Math. Notes 85, No. 3, 348-352 (2009); translation from Mat. Zametki 85, No. 3, 351-355 (2009).
Sufficient conditions for the discreteness of the spectrum are obtained for the matrix Sturm-Liouville problem on the half-line:
\[ Y''+\lambda Q(x)Y=0,\; x\geq 0,\; Y(0)=0, \] where \(Q(x)\) is a \(2\times 2\) symmetric continuous positive matrix-function.
MSC:
34B24 Sturm-Liouville theory
47E05 General theory of ordinary differential operators (should also be assigned at least one other classification number in Section 47-XX)
34L15 Eigenvalues, estimation of eigenvalues, upper and lower bounds of ordinary differential operators
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[1] B.M. Levitan and G. A. Suvorechenkova, ”Sufficient conditions for discreteness of the spectrum of a Sturm-Liouville equation with operator coefficient,” Funktsional. Anal. i Prilozhen. 2(2), 56–62 (1968). · Zbl 0188.46203 · doi:10.1007/BF01075948
[2] R. S. Ismagilov, ”The asymptotic behavior of the spectrum of a differential operator in a space of vector valued functions,” Mat. Zametki 9(6), 667–676 (1971).
[3] R. S. Ismagilov and A. G. Kostyuchenko, ”On the spectrum of the vector Schrödinger operator,” Dokl. Ross. Akad. Nauk 411(4), 449–451 (2006). · Zbl 1208.34140
[4] M. G. Gimadislamov and R. S. Ismagilov, ”discreteness of the spectrum of a Sturm-Liouville operator in the space of vector functions,” in Complex Analysis, Differential Equations, Numerical Methods and Applications (Institute of Mathematics and Computer Center, Russian Academy of Sciences, Ufa, 1996), Vol. 4, pp. 29–36 [in Russian].
[5] I. M. Glazman, Direct Methods of Qualitative Spectral Analysis of Singular Differential Operators (Fizmatgiz, Moscow, 1963) [in Russian].
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