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Density, spectral theory and homoclinics for singular Sturm-Liouville systems. (English) Zbl 0816.34016

The singular system: \(- ({\mathcal P} (t) u')' + {\mathcal Q} (t) u = \lambda {\mathcal R} (t)u\) (where for all \(t \in I \subset \mathbb{R}\) the coefficients \({\mathcal P}\), \({\mathcal Q}\) and \({\mathcal R}\) are symmetric matrices) is considered, and two main results are obtained. Appropriate boundary conditions (depending only on the coefficients of the system) are defined such that the corresponding boundary value problem is equivalent to its weak formulation. These (density) conditions ensure that the domain in the weak formulation is a “normal space of distributions”. Other conditions on the coefficients are given under which the set of eigenfunctions is total and orthogonal. These results include the Hermite, the Laguerre and the Bessel special functions as well as the Legendre and the Chebyshev polynomials. The results are applied also to the problem on the real line, and numerical investigations of an example of this problem are presented.
Reviewer: Y.P.Mishev (Sofia)

MSC:

34B24 Sturm-Liouville theory
34L10 Eigenfunctions, eigenfunction expansions, completeness of eigenfunctions of ordinary differential operators
34B30 Special ordinary differential equations (Mathieu, Hill, Bessel, etc.)
33E30 Other functions coming from differential, difference and integral equations
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