Hostname: page-component-848d4c4894-x24gv Total loading time: 0 Render date: 2024-05-16T19:01:57.128Z Has data issue: false hasContentIssue false
  • English
  • Français

1.—On Square-integrable Solutions of Symmetric Systems of Differential Equations of Arbitrary Order.*

Published online by Cambridge University Press:  14 February 2012

V. I. Kogan  and
F. S. Rofe-Beketov
Get access Rights & Permissions [Opens in a new window]

Synopsis

The following results are obtained for symmetric differential expressions of arbitrary order r ≧ 1 with matrix coefficients on the half-line, with a non-negative (and possibly identically degenerate) weight matrix W(t), and with a spectral parameter λ: upper and lower bounds for the deficiency indices N(λ) are found; it is proved that N(λ) is independent of λ for Im λ< 0and for Im λ>0; under very general conditions it is proved that the maximum possible values of the deficiency indices in the half-planesIm λ≷0 can only be attained simultaneously; sufficient conditions for first-order expressions to be quasi-regular are derived; and it isshown that a symmetric system of any order reduces to a canonical, first-order system.

Examples are constructed, and the case of the whole line is also touched on.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 74 , 1976 , pp. 5 - 40
Copyright
Copyright © Royal Society of Edinburgh 1976

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1Kogan, V. I. and Rofe-Beketov, F. S.. The deficiency indices of differential operators of odd order. Kharkov Polytechn. Inst. named after Lenin. Materials of the Sci. and Techn. Conference on Results of Sci. Work in 1969 7 (1970), 93–95. (Kharkov: State Univ.)Google Scholar
2Everitt, W. N.. Integrable-square solutions of ordinary differential equations. Quart. J. Math Oxford Ser. 10 (1959), 145–155.CrossRefGoogle Scholar
3Everitt, W. N.. Singular differential equations. I. The even-order case. Math. Ann. 156 (1964), 9–24.CrossRefGoogle Scholar
4Everitt, W. N.. Integrable-square, analytic solutions of odd-order, formally symmetric, ordinary differential equations. Proc. London Math. Soc. 25 (1972), 156–182.CrossRefGoogle Scholar
5Atkinson, F. V.. Discrete and continuous problems (New York: Academic, 1964).Google Scholar
6Gokhberg, I. Ts. and Krein, M. G.. The theory of Volterra operators in Hilbert space, and its applications (Moscow, 1967).Google Scholar
7Daletskiy, Yu. L. and Krein, M. G.. The stability of solutions of differential equations in Banach space (Moscow, 1970).Google Scholar
8Riesz, F. and Szökefalvi-Nagy, B.. Lectures on functional analysis (New York: Ungar, 1955).Google Scholar
9Kato, T.. Theory of perturbations of linear operators (Berlin: Springer, 1966).Google Scholar
10Everitt, W. N.. Integrable-square solutions of ordinary differential equations, II. Quart. J. Math. Oxford Ser. 13 (1962), 217–220.CrossRefGoogle Scholar
11Orlov, S. A.. Abstracts of lectures to the All-Union conference of the theory of functions of a complex variable, 165–167 (Kharkov: Akad. Nauk Ukrainian SSR, 1971).Google Scholar
12Orlov, S. A.. Deficiency indices of linear differential operators. Doklady Akad. Nauk SSSR 92 (1953), 483–486.Google Scholar
13Everitt, W. N.. Inequalities for Gram determinants. Quart. J. Math. Oxford Ser. 8 (1957), 191–196.CrossRefGoogle Scholar
14Everitt, W. N.. Some properties of Gram matrices and determinants. Quart. J. Math. Oxford Ser. 9 (1958), 87–98.CrossRefGoogle Scholar
15Gorbachuk, M. L.. The spectral functions of second-order differential equations with operator coefficients. Ukrainian Math. J. 18, 2 (1966),3–21.Google Scholar
16Krein, M. G.. Infinite J-matrices and the matrix problem of moments. Doklady Akad. Nauk SSSR 69 (1949), 125–128.Google Scholar
17Berezanskiy, Yu. M.. Expansions in eigenfunctions of self-adjoint operators. (Transl. Math. Monogr. 17.) Providence: Amer. Math. Soc, 1968.CrossRefGoogle Scholar
18Akhiezer, N. I. and Glazman, I. M.. Theory of linear operators in Hilbert space, 2nd edn. (Moscow, 1966).Google Scholar
19Gokhberg, I. Ts. and Krein, M. G.. Introduction to the theory of non-self adjoint operators in Hilbert space, Moscow, 1965. (Transl. Math. Monogr. 18.) (Providence: Amer. Math. Soc, 1969).Google Scholar
20Hartman, P.. Ordinary differential equations (New York: Wiley, 1964).Google Scholar
21Bellman, R.. Stability theory of differential equations (New York:McGraw-Hill, 1953).Google Scholar
22Cesari, L.. Asymptotic behaviour and stability problems in ordinary differential equations (Berlin: Springer, 1959).Google Scholar
23Zelenko, L. B.. The deficiency indices and spectrum of a self-adjoint, first-order system of differential equations. Doklady Akad. Nauk SSSR 181 (1968), 278–281.Google Scholar
24Naimark, M. A.. Linear differential operators (New York: Ungar, 1968).Google Scholar
25Rofe-Beketov, F. S.. Self-adjoint extensions of differential operators in the space of vectorvalued functions. Doklady Akad. Nauk. SSSR 184 (1969), 1034–1037.Google Scholar
26Rofe-Beketov, F. S.. Self-adjoint extensions of differential operators in the space of vectorvalued functions. Teor. Funkciľ Funkcional. Anal. i Priložen. Vyp. 8 (1969), 3–24.Google Scholar
27Coddington, E. A. and Levinson, N.. Theory of ordinary differential equations (New York: McGraw-Hill, 1954).Google Scholar
28Kogan, V. I.. Operators generated by 7p-matrices in the case of maximal deficiency indices. Teor. Funkciľ Funkcional. Anal. i Priložen. Vyp. 11 (1970), 103–107.Google Scholar
29Bennewitz, C.. Generalization of some statements by Kodaira. Proc. Roy. Soc. Edinburgh Sect. A. 71(1973), 113–119.Google Scholar
30Kogan, V. I. and Rofe-Beketov, F. S.. On the problem of the deficiency indices of symmetric differential operators with complex coefficients. Mathematical Physics and Functional Analysis. Trudy Fiz. Tekhn. Inst. Niz. Temp., Akad. Nauk, Ukrainian SSSR 2 (1971), 45–60.Google Scholar
31Lappo-Danilevskiy, I. A.. Application of functions of matrices to thetheory of linear systems of ordinary differential equations (Moscow, 1957).Google Scholar
32Erugin, N. P.. Lappo-Danilevskiy's method in the theory of linear differential equations (Leningrad State Univ., 1956).Google Scholar
33Levitan, B. M. and Sargsyan, I. S.. Introduction to spectral theory (Moscow, 1970).Google Scholar
34Martinov, V. V.. Conditions for the spectrum of a self-adjoint system offirst-order differential equations to be discrete or continuous. Doklady Akad. Nauk SSSR 165 (1965), 996–999.Google Scholar
35Kalyabin, G. A.. On the number of solutions in L2(0, ∞) of a self-adjoint system of second-order differential equations. Funkcional. Anal, i Priložen 6, 3 (1972), 74–76.Google Scholar
36Lidskiy, V. B.. The number of integrable-square solutions of a system of differential equations. Doklady Akad. Nauk SSSR 95 (1954), 217–220.Google Scholar
37Glazman, I. M.. The deficiency indices of differential operators. Doklady Akad. Nauk SSSR 64 (1949), 151–154.Google Scholar