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Trace Formulas for Powers of a Sturm-Liouville Operator

Published online by Cambridge University Press:  20 November 2018

Richard C. Gilbert  and
Vernon A. Kramer
Richard C. Gilbert
Affiliation:
University of California at Riverside and Mathematics Research Center, University of Wisconsin
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Let H0 be the mth power (m a positive integer) of the self-adjoint operator defined in the Hilbert space L2(0, π) by the differential operator — (d2/dx2) and the boundary conditions u(0) = u(π) = 0. The eigenvalues of H0 are μn = n2m and the corresponding eigenfunctions are ϕn = (2/π)1/2 sin nx, n = 1 , 2 , . . ..

Let p be a (2m — 2)-times continuously differentiate real valued function defined over the interval [0, π] satisfying the conditions p(j)(0) = p(j)(π) = 0 for j odd and less than 2m — 4.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 16 , 1964 , pp. 412 - 422
Copyright
Copyright © Canadian Mathematical Society 1964

References

1. Courant, R. and Hilbert, D., Methods of mathematical physics, vol. I (New York, 1953).Google Scholar
2. Dikii, L. A., Trace formulas for Sturm-Liouville differential operators, Usp. Mat. Nauk, 13 (1958), 111–143 (AMS Translations, Ser. 2, JS ,1961).Google Scholar
3. Kato, T., On the convergence of the perturbation method, J. Fac. Sci., Univ. of Tokyo, Sect. 1, 6, part 3 (1951).Google Scholar
4. Schatten, R., Norm ideals of completely continuous operators, Ergebnisse der Mathematik und ihrer Grenzgebiete, vol. 27 (1960).Google Scholar