The stability of many physical systems depends on spectral properties of ordinary differential operators posed on the entire line. For numerical purposes, one restricts the all-line spectral problem to a finite interval spectral problem. The question of how the two problems are related then arises. In this work, we study principal (or generalized) eigenvalue problems for ordinary differential equations on the infinite line and bounded, but large, intervals by writing them as matrix problems. Matrix formulations allow us to prove that eigenvalue problems on finite intervals are perturbations of the all-line eigenvalue problem if the boundary conditions satisfy a determinant condition. Using this condition, we also prove the convergence of Green's functions of an ordinary differential operator on the infinite line and large bounded intervals for a spectral parameter that is in the resolvent of the operator or a simple eigenvalue of the operator. This work may be interesting for graduate students and researchers focused on stability questions, spectral problems, and scientific computations.

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