Boundary conditions in the infinite interval and some related results.

Govindaraju, Rao V.

doi:10.1090/S0002-9947-1971-0269912-2

Boundary conditions in the infinite interval and some related results.
HTML articles powered by AMS MathViewer

by Rao V. Govindaraju

Trans. Amer. Math. Soc. 154 (1971), 113-128

DOI: https://doi.org/10.1090/S0002-9947-1971-0269912-2

PDF | Request permission

Abstract:

The number of square-integrable solutions of a real, selfadjoint differential equation are determined using exclusively the elementary theory of matrices. Boundary conditions in the infinite interval are given a simple format and a relation between any two selfadjoint boundary conditions is deduced. Finally a lemma due to Titchmarsh, which forms the basis of eigenfunction expansions, is generalized.

References

W. N. Everitt, Singular differential equations. II. Some self-adjoint even order cases, Quart. J. Math. Oxford Ser. (2) 18 (1967), 13–32. MR 209545, DOI 10.1093/qmath/18.1.13
E. C. Titchmarsh, Eigenfunction Expansions Associated with Second-Order Differential Equations, Oxford, at the Clarendon Press, 1946 (German). MR 0019765
Earl A. Coddington and Norman Levinson, Theory of ordinary differential equations, McGraw-Hill Book Co., Inc., New York-Toronto-London, 1955. MR 0069338
R. Courant and D. Hilbert, Methods of mathematical physics. Vol. I, Interscience Publishers, Inc., New York, N.Y., 1953. MR 0065391
W. N. Everitt, Some properties of Gram matrices and determinants, Quart. J. Math. Oxford Ser. (2) 9 (1958), 87–98. MR 96689, DOI 10.1093/qmath/9.1.87
W. N. Everitt, Integrable-square solutions of ordinary differential equations, Quart. J. Math. Oxford Ser. (2) 10 (1959), 145–155. MR 107039, DOI 10.1093/qmath/10.1.145
Kunihiko Kodaira, On ordinary differential equations of any even order and the corresponding eigenfunction expansions, Amer. J. Math. 72 (1950), 502–544. MR 36404, DOI 10.2307/2372051
W. N. Everitt, Fourth order singular differential equations, Math. Ann. 149 (1962/63), 320–340. MR 152693, DOI 10.1007/BF01471126
W. N. Everitt, Singular differential equations. I. The even order case, Math. Ann. 156 (1964), 9–24. MR 166433, DOI 10.1007/BF01359977

On the theory of singular differential operators

5

4

13