Sturm-Liouville Problems

A Sturm-Liouville problem consists of an ordinary differential equation of the form

-(py')'+qy=zwy on (a,b)
and a boundary condition such as
y(a)=y(b)=0.
Here p, q and w are given Lebesgue integrable functions on (a,b) and z is the so-called spectral parameter. The goal of study is to find out for which values of z the equation has non-trivial solutions satisfying the boundary condition. These values of z are called the eigenvalues of the problem and the corresponding non-trivial solutions their eigenfunctions. Such problems have a wide range of applications in engineering and science, as well as in mathematics.

References

  1. Q. Kong, H. Wu & A. Zettl: Dependence of eigenvalues on the problem. Math. Nachr. 188 (1997), 173-201.
  2. M. S. P. Eastham, Q. Kong, H. Wu & A. Zettl: Inequalities among eigenvalues of Sturm-Liouville problems. J. Inequalities Appl. 3 (1999), 25-43.
  3. Q. Kong, H. Wu & A. Zettl: Dependence of the n-th Sturm-Liouville eigenvalues on the problem. J. Differential Equations 156 (1999), 328-354.
  4. Q. Kong, H. Wu & A. Zettl: Inequalities among eigenvalues of singular Sturm-Liouville problems. Dynamic Systems & Appl. 8 (1999), 517-531.
  5. Q. Kong, Q. Lin, H. Wu & A. Zettl: A new proof of the inequalities among Sturm-Liouville eigenvalues. PanAmerican Math. J. 10 (2000), no. 2, 1-10.
  6. Q. Kong, H. Wu & A. Zettl: Geometric aspects of Sturm-Liouville problems, I. Structures on spaces of boundary conditions. Royal Soc. Edinburgh Proc. 130A (2000), 561-589.
  7. Q. Kong, H. Wu & A. Zettl: Left-definite Sturm-Liouville problems. J. Differential Equations 177 (2001), 1-26.
  8. Q. Kong, H. Wu & A. Zettl: Sturm-Liouville problems with finite spectrum. J. Math. Anal. Appl. 263 (2001), 748-762.
  9. K. Haertzen, Q. Kong, H. Wu & A. Zettl: Geometric aspects of Sturm-Liouville problems, II. Space of boundary conditions for left-definiteness. Preprint, 2000 August 03.
  10. X. Cao, Q. Kong, H. Wu & A. Zettl: Sturm-Liouville problems whose leading coefficient function changes sign. Preprint, 2001 August 18.
  11. P. Bailey, J. Billingham, R. Cooper, W. Everitt, A. King, Q. Kong, H. Wu & A. Zettl: Eigenvalue problems in fuel cell dynamics. Proc. Royal Society of London, to appear.
  12. X. Cao, Q. Kong, H. Wu & A. Zettl: Geometric aspects of Sturm-Liouville problems, III. Level surfaces of the n-th eigenvalue. Preprint, 2002.
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Last modified on 2002 October 03. wu@math.niu.edu