Oscillation and spectral properties of self-adjoint even order differential operators with middle terms

O. Dosly, Department of Mathematics, Masaryk University, Brno, Czech Republic

E. J. Qualitative Theory of Diff. Equ., Proc. 7'th Coll. Qualitative Theory of Diff. Equ., No. 7. (2004), pp. 1-21.

Abstract: Oscillation and spectral properties of even order self-adjoint differential operators of the form
$$
L(y):=\frac{1}{w(t)}\sum_{k=0}^n (-1)^k \left(r_k(t)y^{(k)}\right)^{(k)},\quad r_n(t)>0\, ,w(t)>0, %\tag{L}
$$
are investigated. A particular attention is devoted to the fourth order operators with a middle term, for which new (non)oscillation criteria are derived. Some open problems and perspectives of further research are discussed.

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