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On Titchmarsh-Kodaira’s Formula concerning Weyl-Stone’s Eigenfunction Expansion

Published online by Cambridge University Press:  22 January 2016

Kôsaku Yosida
Kôsaku Yosida*
Affiliation:
Mathematical Institute, Nagoya University
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Let q (x) be real and continuous in the infinite open interval (- ∞, ∞) and let y1(x, λ), y2(x, λ) be the solutions of

(1.1)

with the initial conditions

(1.2) .

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 1 , June 1950 , pp. 49 - 58
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1950

References

1) The case of finite or half finite open interval may be treated exactly in the same manner. Moreover (apparently) general equation may be reduced to (1.1) by the Liouville Transformation .

2) Über gewöhnlichc Difïerentialgleichungen mit Singularitäten und die zugehörigen Entwicklungen willkürlichen Funktionen, Math. Ann., 68 (1910), 220-269.

3) Linear transformations in Hilbert space, Amer. Math. Soc. Coll. Publ. XV (1932).

4) Eigenfunction expansisons associated with second order differential equations, Oxford (1946).

5) The eigenvalue problem for ordinary differential equations of the second order and Heisenberg’s theory of S-matrices, Amer. J. of Math., 71 (1949), 921-945.

6) H. Weyl or E. C. Titchmarsh, loc. cit.