Hostname: page-component-78c5997874-v9fdk Total loading time: 0 Render date: 2024-11-19T13:21:33.371Z Has data issue: false hasContentIssue false
  • English
  • Français

On subordinacy and analysis of the spectrum of Schrödinger operators with two singular endpoints

Published online by Cambridge University Press:  14 November 2011

D. J. Gilbert
D. J. Gilbert
Affiliation:
Department of Applied Mathematics, University of Hull, Hull HU6 7RX, U.K.
Get access Rights & Permissions [Opens in a new window]

Synopsis

The theory of subordinacy is extended to all one-dimensional Schrödinger operatorsfor which the corresponding differential expression L = – d2/(dr2) + V(r) is in the limit point case at both ends of an interval (a, b), with V(r) locally integrable. This enables a detailed classification of the absolutely continuous and singular spectra to be established in terms of the relative asymptotic behaviour of solutions of Lu = xu, x εℝ, as r→a and r→b. The result provides a rigorous but straightforward method of direct spectral analysis which has very general application, and somefurther properties of the spectrum are deduced from the underlying theory.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 112 , Issue 3-4 , 1989 , pp. 213 - 229
Copyright
Copyright © Royal Society of Edinburgh 1989

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

1Weyl, H.. Über gewohnliche Differentialgleichungen mit singularitäten und die zugehörigen Entwicklungen willkürlicher Funktionen. Math. Ann. 68 (1910), 220–269.CrossRefGoogle Scholar
2Titchmarsh, E. C.. Eigenfunction Expansions associated with Second Order DifferentialEquations, Vol. I (London/Oxford: Oxford University Press (Clarendon), 1946).Google Scholar
3Coddington, E. A. and Levinson, N.. Theory of Ordinary Differential Equations. (New York: McGraw-Hill, 1955).Google Scholar
4Glazman, I. M.. Direct Methods of Qualitive Spectral Analysis of Singular Differential Operators (Jerusalem: Israel Program for Scientific Translations, 1965).Google Scholar
5Levitan, B. M. and Sarsjan, I. S.. Introduction to Spectral Theory. Amer. Math. Soc. Transl., Vol. 39 (Providence, RI: American Mathematical Society, 1975).Google Scholar
6Kodaira, 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
7Wolfson, K. G.. On the spectrum of a boundary value problem with two singular endpoints. Amer. J. Math. 72 (1950), 713–719.CrossRefGoogle Scholar
8Kac, I. S.. On the multiplicity of the spectrum of a second order differential operator. Izv. Akad. Nauk SSSR Ser. Mat. 27 (1963), 1081–1112 (in Russian).Google Scholar
9Gilbert, D. J. and Pearson, D. B.. On subordinacy and analysis of the spectrum of onedimensional Schrodinger operators. J. Math. Anal. Appl. 128 (1987), 30–56.CrossRefGoogle Scholar
10Stone, M. H.. Linear Transformations in Hilbert Space. Amer. Math. Soc. Colloq. Publ., Vol. XV (New York: American Mathematical Society, 1932).Google Scholar
11Gilbert, D. J.. Subordinacy and Spectral Analysis of Schrödinger Operators (Ph.D. Thesis, University of Hull, 1984).Google Scholar
12Dunford, N. and Schwartz, J. T.. Linear Operators, Part II (New York: Interscience, 1963).Google Scholar
13Plessner, A.. Über das Verhalten analytischer Funktionen am Rande ihres Definitionsbereichs. J. Reine Angew. Math. 158 (1927), 219–227.CrossRefGoogle Scholar
14Lusin, N. N. and Privalov, I. I.. Sur l'unicité et la multiplicité des fonctions analytiques. Ann. Sci. École Norm. Sup. (3) 42 (1925), 143–191.CrossRefGoogle Scholar
15Dolzhenko, E. P. and Tumarkin, G. Ts.. Lusin, N. N.and the theory of boundary propertiesof analytic functions. Russian Math. Surveys 40, 3 (1985), 79–95.CrossRefGoogle Scholar
16Noshiro, K.. Cluster Sets (Berlin: Springer-Verlag, 1960).CrossRefGoogle Scholar
17Kato, T.. On finite-dimensional perturbations of self-adjoint operators. J. Math. Soc. Japan 9 (1957), 239–249.CrossRefGoogle Scholar
18Weidmann, J.. Spectral Theory of Ordinary Differential Operators (Berlin: Springer-Verlag, 1987).CrossRefGoogle Scholar
19Plesner, A. I.. Spectral Theory of Ordinary Differential Operators, Vol. II (NewYork: F. Ungar, 1969)Google Scholar
20Donoghue, W. F. Jr, Monotone Matrix Functions and Analytic Continuation (Berlin: Springer Verlag, 1974).CrossRefGoogle Scholar
21Saks, S.. Theory of the Integral, 2nd edn (New York: Hafner, 1937).Google Scholar
22Blokh, A. Sh.. On the determination of a differential equation by its spectral matrix function. Dokl. Akad. Nauk SSSR 92 (1953), 209–212 (in Russian).Google Scholar
23Naimark, M. A.. Linear Differential Operators, Part II (London: Harrap, 1968).Google Scholar
24Hille, E.. Lectures on Ordinary Differential Equations (Massachusetts: Addison-Wesley, 1969).Google Scholar
25Choudhuri, J. and Everitt, W.. On the spectrum of ordinary second order differential operators. Proc. Roy. Soc. Edinburgh Sect. A 68 (1968), 95–119.Google Scholar
26Kac, I. S.. On the multiplicity of the spectrum of a second order differential operator. Soviet Math. 3 (1962), 1035–1039.Google Scholar
27Combes, J. M., Duclos, P. and Seiler, R., Krein's formula and one-dimensional multiple well. J. Fund. Anal. 52 (1983), 257–301.CrossRefGoogle Scholar
28Behncke, H.. Finite-dimensional perturbations. Proc. Amer. Math. Soc. 72 (1978), 82–84.Google Scholar
29Hartman, P. and Wintner, A.. Oscillatory and non-oscillatory linear differential equations. Amer. J. Math. 71 (1949) 627–649.CrossRefGoogle Scholar
30Merzbacher, E.. Quantum Mechanics, 2nd edn (New York: Wiley, 1970).Google Scholar
31Weidmann, J.. Linear Operators in Hilbert Space (Berlin: Springer-Verlag, 1980).CrossRefGoogle Scholar
32Hartman, P.. A characterisation of the spectra of one-dimensional wave equations. Amer. J. Math. 71 (1949), 915–920.CrossRefGoogle Scholar
33Barra, G. de. Measure Theory and Integration (Chichester: Ellis Norwood, 1981).Google Scholar
34Gel'fand, I. M. and Levitan, B.. On the determination of a differential equation from its spectral function. Amer. Math. Soc. Transl. 1 (1955), 253–304.Google Scholar
35Eastham, M. S. P. and Kalf, H.. Schrödinger-type Operators with Continuous Spectra (Boston: Pitman, 1982).Google Scholar
36Hartman, P. and Wintner, A.. A separation theorem for continuous spectra. Amer. J. Math. 71 (1949), 650–662.CrossRefGoogle Scholar
37Aronszajn, N. A.. On a problem of Weyl in the theory of singular Sturm-Liouville equations. Amer. J. Math. 79 (1957), 597–610.CrossRefGoogle Scholar