The Laplace Transform and Spectral Representations of Polynomial Operator Pencils
Abstract
The spectral representation associated with the polynomial operator pencil L_{0} + λ L_{1} + λ ^{2}L_{2} + ldots + λ ^{N}L_{N}, where L_{n} (n = 0, 1, 2, ldots , N) are linear operators and λ is a complex parameter, is derived formally using the Laplace transform. The derivation involves a conversion of the eigenvalue problem for the operator pencil into an initialvalue problem by replacing λ with partial /partial t and introducing N1 initial conditions. This procedure yields the spectral representation in the form of an inverse Laplace transform of the Green's operator associated with the operator pencil. The results of this paper are illustrated with examples and provide a simple but powerful and systematic approach to nonstandard eigenvalue problems for linear operators. These examples are a 2 × 2 matrix problem which has three eigenvalues, a SturmLiouvilleRossby type wave equation discussed recently by A. Masuda (Q. appl. Math. 47, 435445 (1989)), and a classical problem in which the eigenvalue parameter appears not only in the differential equation but also in the boundary conditions.
 Publication:

Proceedings of the Royal Society of London Series A
 Pub Date:
 November 1993
 DOI:
 10.1098/rspa.1993.0149
 Bibcode:
 1993RSPSA.443..333I