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Determination of Weyl's m-coefficient for a continuous spectrum

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Ordinary and Partial Differential Equations

Part of the book series: Lecture Notes in Mathematics ((LNM,volume 415))

Abstract

The spectral properties of a self-adjoint second order operator of the limit-point type are discussed for the case of a continuous spectrum. A method that employs real numerical integration of the initial solutions in combination with the knowledge of a fundamental system of exponential solutions is presented. The latter are conveniently calculated from Riccati's differential equation.

The method is applied to the Stark effect in the hydrogen atom and agreement with previous results based on Airy functions in the asymptotic region is found.

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5. References

  1. H. Weyl, Math. Ann. 63, 220 (1910).

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  2. E. C. Titchmarsh, Eigenfunction expansions associated with second order differential equations, Oxford University Press. Part I; ibid, Part II (1946,1958).

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  3. J. Chaudhuri and W. N. Everitt, Proc. Royal Soc. Edinburgh (A) 68, 95 (1968).

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  4. E. C. Titchmarsh, Proc. Roy. Soc. A 207, 321 (1951).

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  5. M. Hehenberger, H. V. McIntosh and E. Brändas, Preliminary Research Report No 380, 1973. Quantum Chemistry Group, Uppsala University. (to be published in Phys. Rev.)

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  6. E. Brändas, M. Hehenberger and H. V. McIntosh, Preliminary Research Report No 379, 1973. Quantum Chemistry Group, Uppsala University. (to be published in Int. J. Quant. Chem.)

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  7. R. Jost, Physica 12, 509 (1946).

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Authors
  1. Erkki Brändas
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  2. Michael Hehenberger
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Editor information

B. D. Sleeman I. M. Michael

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© 1974 Springer-Verlag

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Brändas, E., Hehenberger, M. (1974). Determination of Weyl's m-coefficient for a continuous spectrum. In: Sleeman, B.D., Michael, I.M. (eds) Ordinary and Partial Differential Equations. Lecture Notes in Mathematics, vol 415. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0065542

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  • DOI: https://doi.org/10.1007/BFb0065542

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-06959-1

  • Online ISBN: 978-3-540-37264-6

  • eBook Packages: Springer Book Archive

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