Premium
Connection Formulas for Second Order Differential Equations with a Complex Parameter and Having an Arbitrary Number of Turning Points
Author(s) -
Eberhard W.,
Freiling G.,
Schneider A.
Publication year - 1994
Publication title -
mathematische nachrichten
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.913
H-Index - 50
eISSN - 1522-2616
pISSN - 0025-584X
DOI - 10.1002/mana.19941650114
Subject(s) - mathematics , connection (principal bundle) , eigenvalues and eigenvectors , interval (graph theory) , order (exchange) , turning point , differential equation , mathematical analysis , point (geometry) , boundary value problem , differential (mechanical device) , boundary (topology) , combinatorics , geometry , period (music) , physics , finance , quantum mechanics , acoustics , engineering , economics , aerospace engineering
We consider the differential equation\documentclass{article}\pagestyle{empty}\begin{document}$ - u '' + \chi (x)u = \rho ^2 \phi ^2 (x)u $\end{document} on a finite interval I, where I contains m turning points, that is here, zeros of ϕ. Using asymptotic estimates proved by R. E. LANGER for solutions of (*) for intervals containing only one turning point we derive asymptotic estimates (for ρ → ) 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.