Discreteness of some continuous spectrum eigenfunction expansions

Authors:
Don B. Hinton and Robert M. Kauffman

Journal:
Proc. Amer. Math. Soc. **119** (1993), 235-243

MSC:
Primary 34L10; Secondary 47A70, 47E05

DOI:
https://doi.org/10.1090/S0002-9939-1993-1174493-5

MathSciNet review:
1174493

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We discuss replacing the integrals in continuous spectrum expansions by finite sums, in the special case of the Dirichlet problem for second-order ordinary differential operators on a half-line. The error is controlled in the operator norm of , where and are natural Hilbert spaces for the problem.

**[1]**F. V. Atkinson,*On the asymptotic behavior of the Titchmarsh-Weyl*-*coefficient and the spectral function for scalar second-order differential expressions*, Lecture Notes in Math., vol. 964, Springer-Verlag, Berlin, Heidelberg, and New York, 1982, pp. 1-27. MR**693099 (85b:34022)****[2]**K. Chadan and P. C. Sabatier,*Inverse problems in quantum scattering theory*, Springer-Verlag, Berlin, Heidelberg, and New York, 1989. MR**985100 (90b:81002)****[3]**E. A. Coddington and N. Levinson,*Theory of ordinary differential equations*, McGraw-Hill, New York, 1965. MR**0069338 (16:1022b)****[4]**W. N. Everitt, D. B. Hinton, and J. S. W. Wong,*On the strong limit*-*classification of linear ordinary differential expressions of order*, Proc. London Math. Soc. (3)**29**(1974), 351-357. MR**0409956 (53:13708)****[5]**D. B. Hinton and J. K. Shaw,*Absolutely continuous spectra of second order differential operators with short and long range potentials*, SIAM J. Math. Anal.**17**(1986), 182-196. MR**819222 (87c:34038)****[6]**D. B. Hinton, M. Klaus, and J. K. Shaw,*Series representation and asymptotics for Titchmarsh-Weyl**functions*, Differential Integral Equations**2**(1989), 419-429. MR**996750 (90m:34057)****[7]**R. M. Kauffman,*Finite eigenfunction approximation for continuous spectrum operators*, Internat. J. Math. Math. Sci. (to appear). MR**1200106 (94e:47030)****[8]**M. Klaus,*On the variation-diminishing property of Schrödinger operators*, Oscillation, Bifurcation, and Chaos, CMS Conference Proceedings, vol. 8, Amer. Math. Soc., Providence, RI, 1986, pp. 199-212. MR**909910 (88m:34026)****[9]**V. A. Marchenko,*Sturm-Liouville operators and applications*, Birkhäuser Verlag, Boston, MA, 1986. MR**897106 (88f:34034)****[10]**E. C. Titchmarsh,*Eigenfunction expansions*, Part 1, Oxford Univ. Press, Oxford, 1962. MR**0176151 (31:426)**

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC:
34L10,
47A70,
47E05

Retrieve articles in all journals with MSC: 34L10, 47A70, 47E05

Additional Information

DOI:
https://doi.org/10.1090/S0002-9939-1993-1174493-5

Article copyright:
© Copyright 1993
American Mathematical Society