Hostname: page-component-cd9895bd7-fscjk Total loading time: 0 Render date: 2024-12-26T06:49:37.721Z Has data issue: false hasContentIssue false
  • English
  • Français

Singular eigenvalue problems with eigenvalue parameter contained in the boundary conditions

Published online by Cambridge University Press:  14 November 2011

Charles T. Fulton
Charles T. Fulton
Affiliation:
The Pennsylvania State University, University Park, Pennsylvania 16802, U.S.A.
Get access Rights & Permissions [Opens in a new window]

Synopsis

In this paper I extend the analysis of regular problems containing the eigenvalue parameter in the boundary conditions given by Walter (1973) and myself (1977) to singular problems which involve the eigenvalue parameter linearly in a regular or a limit-circle boundary condition at the left endpoint. The formulation of the limit-circle boundary conditions follows that given in another paper by the present author in 1977, and has the advantage that a λ-dependent boundary condition at a regular endpoint becomes a special case of a λ-dependent boundary condition at a limit-circle endpoint. The simplicity of the spectrum is also built into the formulation given, and the spectral function is shown to have bounded total variation over (−∞, ∞) which is known in terms of the parameters of the λ-dependent boundary condition independently of the limit-circle/limit-point classification at the right endpoint. The theory is applied to the constant coefficient equation in [0, ∞) and the Bessel equation of order zero in (0, ∞), explicit formulae for the spectral function being obtained in each case. Finally, the question is posed as to whether the classical Weyl theory for problems not involving λ in the boundary conditions can also be formulated so as to involve spectral functions having bounded total variation.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 87 , Issue 1-2 , 1980 , pp. 1 - 34
Copyright
Copyright © Royal Society of Edinburgh 1980

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

1Akhiezer, N. I. and Glasmann, I. M.. Theorie der linearen Operatoren im Hilbert Raum (5th edn) (Berlin: Akademie Verlag, 1968).Google Scholar
2Atkinson, F. V.. Discrete and Continuous Boundary Problems (New York: Academic Press, 1964).Google Scholar
3Breiman, L.. Probability (Reading, Mass.: Addison-Wesley, 1968).Google Scholar
4Bognar, J.. Indefinite inner product spaces (New York: Springer, 1974).CrossRefGoogle Scholar
5Butzer, P. L. and Berens, H.. Semi-Groups of operators and approximation (Berlin: Springer, 1967).CrossRefGoogle Scholar
6Coddington, E. A.. Self-adjoint subspace extensions of non-densely defined symmetric operators. Advances in Math. 14 (1974), 309–332.CrossRefGoogle Scholar
7Coddington, E. A.. Self-adjoint problems for nondensely defined ordinary differential operators and their eigenfunction expansions. Advances in Math. 15 (1975), 1–40.CrossRefGoogle Scholar
8Coddington, E. A. and Dijksma, A.. Self-adjoint subspaces and eigenfunction expansions for ordinary differential subspaces. J. Differential Equations 20 (1976), 473–526.CrossRefGoogle Scholar
9Coddington, E. A. and Levinson, N.. Theory of ordinary differential equations (New York: McGraw-Hill, 1955).Google Scholar
10Dijksma, A. and de Snoo, H. S. V.. Eigenfunction expansions for nondensely defined differential operators. J. Differential Equations 17 (1975), 198–219.CrossRefGoogle Scholar
11Dijksma, A. and de Snoo, H. S. V.. Self-adjoint extensions of symmetric subspaces. Pacific J. Math. 54 (1974), 71–100.CrossRefGoogle Scholar
12Dijksma, A. and de Snoo, H. S. V.. Integral transforms and a class of singular S-hermitian eigenvalue problems. Manuscripta Math. 10 (1973), 129–139.CrossRefGoogle Scholar
13Cohen, D. S.. An integral transform associated with boundary conditions containing an eigenvalue parameter. SIAM J. Appl. Math. 14 (1966), 1164–1175.CrossRefGoogle Scholar
14Donoghue, W. F. Jr., Monotone matrix functions and analytic continuation (New York: Springer, 1974).CrossRefGoogle Scholar
15Dunford, N. and Schwartz, J. T.. Linear operators II (New York: Interscience, 1963).Google Scholar
16Everitt, W. N.. On a property of the m-coefficient of a second-order linear differential equation. J. London Math. Soc. 4 (1972), 443–457.CrossRefGoogle Scholar
17Feller, W.. The parabolic differential equations and the associated semigroups of transforms. Ann. of Math. 55 (1952), 468–519.CrossRefGoogle Scholar
18Feller, W.. Generalized second-order differential operators and their lateral conditions. Illinois J. Math. 1 (1957), 459–504.CrossRefGoogle Scholar
19Feller, W.. On differential operators and boundary conditions. Comm. Pure Appl. Math. 8 (1955), 203–216.CrossRefGoogle Scholar
20Feller, W.. Diffusion processes in one dimension. Trans. Amer. Math. Soc. 77 (1954), 1–31.CrossRefGoogle Scholar
21Feller, W.. Two singular diffusion problems. Ann. of Math. 54 (1951), 173–182.CrossRefGoogle Scholar
22Feller, W.. An Introduction to Probability Theory and its Applications II (New York: Wiley, 1966).Google Scholar
23Fine, H. B.. College Algebra (New York: Ginn, 1904).Google Scholar
24Fulton, C.. Parametrizations of Titchmarsh's m(λ)-functions in the limit circle case. Trans. Amer. Math. Soc. 229 (1977), 51–63.Google Scholar
25Fulton, C.. Parametrizations of Titchmarsh's m(λ)-functions in the limit circle case. (R. W. T. H. Aachen: Dissertation, 1973).Google Scholar
26Fulton, C.. Two-point boundary value problems with eigenvalue parameter contained in the boundary conditions. Proc. Roy. Soc. Edinburgh Sect. A 77 (1977), 293–308.CrossRefGoogle Scholar
27Fulton, C. and Pruess, S.. Numerical methods for a singular eigenvalue problem with eigenparame-ter in the boundary conditions. J. Math. Anal. Appl. 71 (1979), 431–462.CrossRefGoogle Scholar
28Glazman, I. M.. Direct methods of qualitative spectral analysis of singular differential operators (Engl. transl.) (Jerusalem: Monson, 1965).Google Scholar
29Hartman, P.. Differential equations with nonoscillatory eigenfunctions. Duke Math. J. 15 (1948), 697–709.CrossRefGoogle Scholar
30Hellwig, G.. Differential operators of mathematical physics (Engl. transl.) (Reading, Mass.: Addison-Wesley, 1967).Google Scholar
31Hellwig, G.. Anfangs- und Randwertprobleme bei partiellen Differentialgleichungen von wechselndem Typus auf den Rändern. Math. Z. 58 (1953), 337–357.CrossRefGoogle Scholar
32Hellwig, G.. Über die Anwendung der Laplace-transformation auf Randwert-probleme. Math. Z. 66 (1957), 371–388.CrossRefGoogle Scholar
33Hellwig, G.. Über die Anwendung der Laplace-transformation auf Ausgleichsprobleme. Math. Nachr. 18 (1958), 281–291.CrossRefGoogle Scholar
34Hille, E.. Lectures on ordinary differential equations (Reading, Mass: Addison-Wesley, 1969).Google Scholar
35Jörgens, K. and Rellich, F.. Eigenwerttheorie gewöhnlicher Differentialgleichungen (Berlin: Springer, 1976).CrossRefGoogle Scholar
36Kac, I. S. and Krein, M. G.. R-Functions-Analytic functions mapping the upper halfplane into itself. Amer. Math. Soc. Transl. (2) 103 (1974), 1–18 (Engl. transl, of Supplement I of the Russian translation of reference [2], Moscow, 1968).Google Scholar
37Kodaira, K.. The eigenvalue problem for ordinary differential equations of the second order and Heisenberg's theory of S-matrices. Amer. J. Math. 71 (1949), 921–945.CrossRefGoogle Scholar
38Krall, A.. Linear methods of applied analysis (Reading, Mass.: Addison-Wesley, 1973).Google Scholar
39Langer, R. E.. A problem in diffusion or in the flow of heat for a solid in contact with a fluid. Tôhoicu Math. J. 35 (1932), 360–375.Google Scholar
40Levinson, N.. A simplified proof of the expansion theorem for singular second order linear differential equations. Duke Math. J. 18 (1951), 57–71.Google Scholar
41Levitan, B. M.. Expansion in characteristic functions of differential equations of the second order (Moscow: GITTL, 1950 (Russian)).Google Scholar
42Levitan, B. M. and Sargsjan, I. S.. Introduction to spectral theory: Self-adjoint ordinary differential operators (Providence, R. I.: A. M. S. Transl., 1975).CrossRefGoogle Scholar
43Levitan, B. M.. On the asymptotic behaviour of the spectral function of a self-adjoint differential equation of the second order. Izv. Akad. Nauk SSSR Ser. Mat. 16 (1952), 325–352; Engl. transl., Amer. Math. Soc. Transl. (2) 101 (1973), 192–221.Google Scholar
44Levitan, B. M.. On the asymptotic behaviour of the spectral function of a self-adjoint differential equation of the second order and on eigenfunction expansions, II. Izv. Akad. Nauk SSSR Ser. Mat 19 (1955), 33–58; Engl. transl., Amer. Math. Soc. Transl (2) 110 (1977), 165–188.Google Scholar
45Loeve, M.. Probability theory (2nd edn) (Princeton: Van Nostrand, 1960).Google Scholar
46Naimark, M. A.. Linear differential operators II (New York: Ungar, 1968).Google Scholar
47Neumann, F.. On a problem of transformations between limit-circle and limit-point differential equations. Proc. Roy. Soc. Edinburgh Sect. A 72 (1975), 187–193.CrossRefGoogle Scholar
48Niessen, H. D.. Singulare S-hermitesche Rand-eigenwertprobleme. Manuscripta Math. 3 (1970), 35–68.CrossRefGoogle Scholar
49Niessen, H. D.. Zum verallgemeinerten zweiten Weylschen Satz. Arch. Math. 22 (1971), 648–656.CrossRefGoogle Scholar
50Niessen, H. D. and Schneider, A.. Integraltransformationen zu singulären S-Hermiteschen Randeigenwertproblemen. Manuscripta Math. 5 (1971), 133–145.CrossRefGoogle Scholar
51Niessen, H. D.. Greensche matrix und die Formel von Titchmarsh-Kodaira für singulare S-hermitesche Eigenwertprobleme. J. Reine Angew. Math. 261 (1973), 164–193.Google Scholar
52Odhnoff, Jan. Operators generated by differential problems with eigenvalue parameter in equation and boundary condition. Meddn Lunds Univ. Mat. Semin. 14 (1959).Google Scholar
53Pleijel, A.. A survey of spectral theory for pairs of ordinary differential operators. Lecture Notes in Mathematics 448, 256–272 (Berlin: Springer, 1975).Google Scholar
54Rellich, F.. Halbbeschränkte gewöhnliche Differentialoperatoren zweiter Ordnung. Math. Ann. 122 (1951), 343–368.CrossRefGoogle Scholar
55Schneider, A.. Eine Bemerkung zum Weyl-Stoneschen Eigenwertproblem. Arch. Math. 17 (1966), 352–358.CrossRefGoogle Scholar
56Schneider, A.. Untersuchungen über singulare reelle S-hermitesche Differentialgleichungssysteme im Normalfall. Math. Z. 107 (1968), 271–296.CrossRefGoogle Scholar
57Schneider, A.. Weitere Untersuchungen über singulare reelle S-hermitesche Differentialgleichungssysteme im Normalfall. Math. Z. 109 (1969), 153–168.CrossRefGoogle Scholar
58Schneider, A.. Die Greensche matrix S-hermitescher Rand-eigenwertprobleme im Normalfall. Math. Ann. 180 (1969), 307–312.CrossRefGoogle Scholar
59Schneider, A.. Zum Entwicklungssatz bei reellen singulären Differentialgleichungs-systemen. Arch. Math. 21 (1970), 192–197.CrossRefGoogle Scholar
60Schneider, A.. A note on eigenvalue problems with eigenvalue parameter in the boundary conditions. Math. Z. 136 (1974), 163–167.CrossRefGoogle Scholar
61Scott, W. F.. On eigenfunction expansions of second-order ordinary differential equations. J. London Math. Soc. 4 (1972), 551–559.CrossRefGoogle Scholar
62Sears, D. B.. Integral transforms and eigenfunction theory. Quart. J. Math. Oxford 5 (1954), 47–58.CrossRefGoogle Scholar
63Titchmarsh, E. C.. Eigenfunction expansion associated with second order differential equations I (2nd edn) (London: Oxford Univ. Press, 1962).CrossRefGoogle Scholar
64Titchmarsh, E. C.. Eigenfunction expansions associated with second order differential equations II (London: Oxford Univ. Press, 1958).Google Scholar
65Walter, J.. Regular eigenvalue problems with eigenvalue parameter in the boundary conditions. Math. Z. 133 (1973), 301–312.CrossRefGoogle Scholar
66Weidmann, J.. Lineare Operatoren in Hilberträumen (Stuttgart: Teubner, 1976).Google Scholar
67Weyl, H.. Über gewöhnliche lineare Differentialgleichungen mit Singulären Stellen und ihre Eigenfunktionen. Nachr. Akad. Wiss. Göttingen Math.-Phys. Kl. II (1910), 442–467.Google Scholar
68Widder, D. V.. The Laplace Transform (2nd edn) (Princeton: Univ. Press, 1946).Google Scholar
69Wray, S. D.. On Weyl's Function m(λ). Proc. Roy. Soc. Edinburgh. Sect. A 74 (1976), 41–48.CrossRefGoogle Scholar
70Yosida, K.. On Titchmarsh-Kodaira's formula concerning Weyl-Stone's eigenfunction expansion. Nagoya Math. J. 1 (1950), 49–58.CrossRefGoogle Scholar
71Yosida, K.. Lectures on differential and integral equations (New York: Interscience, 1960).Google Scholar
72Zecca, P.. Su un problema al contorno per l'equazione Δu + λu = 0. Rend. Accad. Sci. Fis. Mat. Napoli 33 (1966), 279–303.Google Scholar