The Spectra of Differential Operators

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.