The Spectra of Differential Operators
Bounded and Unbounded Domains
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The stability of many physical systems depends onspectral properties of ordinary differentialoperators posed on the entire line. For numericalpurposes, one restricts the all-line spectral problemto a finite interval spectral problem. The questionof how the two problems are related then arises. Inthis work, we study principal (or generalized)eigenvalue problems for ordinary differentialequations on the infinite line and bounded, butlarge, intervals by writing them as matrix problems.Matrix formulations allow us to prove that eigenvalueproblems on finite intervals are perturbations of theall-lin...
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.
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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