Regularization of singular Sturm-Liouville equations. (English) Zbl 1221.47083

Goriunov, Andrii; Mikhailets, Vladimir

Methods Funct. Anal. Topol. 16, No. 2, 120-130 (2010).

The paper is devoted to differential expressions of the form \(l(y) = -(py')'+qy\) on a finite interval, under the following assumptions regarding the coefficients: \(q=Q'\), \(1/p, Q/p, Q^2/p\in L^1\), where the derivative is understood in the distribution sense. The authors’ idea is to represent the equations and operators related to the above expression via first order systems. The latter contain not the distribution \(q\), but a more regular function \(Q\), and such systems can be treated using appropriate quasi-derivatives. In particular, the authors study the dependence of the resulting operators on a parameter, and construct boundary triplets which lead to the descriptions of various classes of extensions of the minimal operators. Some results were known for the case where \(p = 1\) [A. M. Savchuk and A. A. Shkalikov, Math. Notes 66, No. 6, 741–753 (1999); translation from Mat. Zametki 66, No. 6, 897–912 (1999; Zbl 0968.34072)], but under stronger assumptions, others are new for this case, too.

Reviewer: Anatoly N. Kochubei (Kyïv)

MSC:

47E05	General theory of ordinary differential operators
34B08	Parameter dependent boundary value problems for ordinary differential equations
34L40	Particular ordinary differential operators (Dirac, one-dimensional Schrödinger, etc.)
47B25	Linear symmetric and selfadjoint operators (unbounded)

Keywords:

quasi-derivative; boundary triplet; singular Sturm-Liouville operator; regularisation

Citations:

Zbl 0968.34072