Spectral analysis of the integral operator arising from the beam deflection problem on elastic foundation II: eigenvalues

摘要

We analyze the eigenstructure of the integral operator K l , α , k which arise naturally from the beam deflection equation on linear elastic foundation with finite beam. We show that K l , α , k has countably infinite number of positive eigenvalues approaching 0 as the limit, and give explicit upper and lower bounds on each of them. Consequently, we obtain explicit upper and lower bounds on the L 2 -norm of the operator K l , α , k . We also present precise approximations of the eigenvalues as they approach the limit 0, which describes the almost regular structure of the spectrum of K l , α , k . Additionally, we analyze the dependence of the eigenvalues, including the L 2 -norm of K l , α , k , on the intrinsic length L = 2 l α of the beam, and show that each eigenvalue is continuous and strictly increasing with respect to L. In particular, we show that the respective limits of each eigenvalue as L goes to 0 and infinity are 0 and 1 / k , where k is the linear spring constant of the given elastic foundation. Using Newton’s method, we also compute explicitly numerical values of the eigenvalues, including the L 2 -norm of K l , α , k , corresponding to various values of L. MSC: 34L15, 47G10, 74K10.