We consider the differential equation (*I -u"+ x ( x ) u = p 2 f p ( x ) u on a finite interval I, where I contains m turning points, that is here, zeros of 9. Using asymptotic estimates proved by R. E. LANGER for solutions of (*) for intervals containing only one turning point we derive asymptotic estimates (for p -+ co) for a special fundamental system of solutions of (*) in I. The results obtained are fundamental for the investigation of eigenvalue problems defined by (*) and suitable boundary conditions.
Key words Singular Sturm-Liouville problems, Birkhoff-regular problems, asymptotic behavior of eigenvalues, eigenfunction expansion MSC (2000) 34B25, 34L05, 34L10, 34L20
Dedicated to our esteemed colleague F. V. AtkinsonSingular boundary conditions are formulated for non-selfadjoint Sturm-Liouville problems which are limitcircle in a very general sense. The characteristic determinant is constructed and it is shown that it can be used to extend the Birkhoff theory for so called "Birkhoff regular boundary conditions" to the singular case. This is illustrated for a class of singular Birkhoff-regular problems; in particular we prove for this class an asymptotic formula for the eigenvalues and an expansion theorem.
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