Hostname: page-component-586b7cd67f-g8jcs Total loading time: 0 Render date: 2024-11-28T09:53:02.208Z Has data issue: false hasContentIssue false
  • English
  • Français

On subordinacy and spectral multiplicity for a class of singular differential operators

Published online by Cambridge University Press:  14 November 2011

D. J. Gilbert
D. J. Gilbert
Affiliation:
School of Science and Mathematics, Sheffield Hallam University, Sheffield SI 1WB, U.K.
Get access Rights & Permissions [Opens in a new window]

Abstract

The spectral multiplicity of self-adjoint operators H associated with singular differential expressions of the form

is investigated. Based on earlier work of I. S. Kac and recent results on subordinacy, complete sets of necessary and sufficient conditions for the spectral multiplicity to be one or two are established in terms of: (i) the boundary behaviour of Titchmarsh–Weyl m-functions, and (ii) the asymptotic properties of solutions of Lu = λu, λ∈ℝ, at the endpoints a and b. In particular, it is shown that H has multiplicity two if and only if L is in the limit point case at both a and b and the set of all λ for which no solution of Lu = λu is subordinate at either a or b has positive Lebesgue measure. The results are completely general, subject only to minimal restrictions on the coefficients p(r), q(r)and w(r), and the assumption of separated boundary conditions when L is in the limit circle case at both endpoints.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 128 , Issue 3 , 1998 , pp. 549 - 584
Copyright
Copyright © Royal Society of Edinburgh 1998

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 Glazman, I. M.. Theory of Linear Operators in Hilbert Space (London: Pitman, 1981; translated from Russian).Google Scholar
2Avron, J. and Simon, B.. Transient and recurrent spectrum. J. Fund. Anal. 43 (1981), 1–31.Google Scholar
3Behncke, H.. Absolute continuity of Hamiltonians with von Neumann Wigner potentials. Proc. Amer. Math. Soc. 111 (1991), 373–84.Google Scholar
4Brown, A.. A version of multiplicity theory. Math. Surveys 13 (1974), 129–60.Google Scholar
5Choudhuri, J. and Everitt, W. N.. On the spectrum of ordinary second order differential operators. Proc. Roy. Soc. Edinburgh Sect. A 68 (1968), 95–119.Google Scholar
6Clark, S. and Hinton, D.. Strong non-subordinacy and absolutely continuous spectra for Sturm–Liouville equations. Differential Integral Equations 6 (1993), 537–86.Google Scholar
7Coddington, E. A. and Levinson, N.. Theory of Ordinary Differential Equations (New York: McGraw-Hill, 1955).Google Scholar
8Davies, E. B. and Simon, B.. Scattering theory for systems with different spatial asymptotics on the right and left. Comm. Math. Phys. 63 (1978), 277–301.Google Scholar
9Deift, P. and Simon, B.. Almost periodic Schrödinger operators, III: The absolutely continuous spectrum in one dimension. Comm. Math. Phys. 90 (1983), 389–411.Google Scholar
10Dollard, J. D. and Friedman, C. N.. Existence of the Moller wave operators. Ann. Phys. 111 (1978), 251–66.Google Scholar
11Donoghue, W. F. Jr., Monotone Matrix Functions and Analytic Continuation (New York: Springer, 1974).Google Scholar
12Dunford, N. and Schwarz, J. T.. Linear Operators, Part II, Spectral Theory (New York: Interscience, 1963).Google Scholar
13Eastham, M. S. P.. The Spectral Theory of Periodic Differential Equations (Edinburgh: Scottish Academic Press, 1973).Google Scholar
14Eastham, M. S. P. and Kalf, H.. Schrödinger Type Operators with Continuous Spectra, Research Notes in Mathematics 65 (London: Pitman, 1982).Google Scholar
15Fulton, C. T.. Parametrisation of Titchmarsh's m(λ)-functions in the limit circle case. Trans. Amer. Math. Soc. 229 (1977), 51–63.Google Scholar
16Gel'fand, I. M. and Levitan, B. M.. On the determination of a differential equation from its spectral function. Amer. Math. Soc. Transl. Ser. 2 1 (1955), 253–304.Google Scholar
17Gilbert, D. J.. Subordinacy and Spectral Analysis of Schrödinger Operators (PhD thesis, University of Hull, 1984).Google Scholar
18Gilbert, D. J.. On subordinacy and analysis of the spectrum of Schrödinger operators with two singular endpoints. Proc. Roy. Soc. Edinburgh Sect. A 112 (1989), 213–29.Google Scholar
19Gilbert, D. J. and Pearson, D. B.. On subordinacy and analysis of the spectrum of one-dimensional Schrödinger operators. J. Math. Anal. Appl. 128 (1987), 30–56.Google Scholar
20Glazman, I. M.. Direct Methods of Qualitative Spectral Analysis of Singular Differential Operators (Jerusalem: Israel Program for Scientific Translations, 1965).Google Scholar
21Harris, W. A. and Lutz, D. A.. Asymptotic integration of adiabatic oscillators. J. Math. Anal. Appl. 51 (1975), 76–93.Google Scholar
22Hartman, P. and Wintner, A.. Oscillatory and non-oscillatory linear differential equations. Amer. J. Math. 71 (1949), 627–49.Google Scholar
23Hinton, D. B. and Shaw, J. K.. On the absolutely continuous spectrum of the perturbed Hill's equation. Proc. London Math. Soc. (3) 50 (1985), 175–92.Google Scholar
24Itatsu, S. and Kaneta, H.. Spectral matrices for first and second order selfadjoint ordinary differential operators with long range potentials. Funkcial. Ekvac. 24 (1981), 23–45.Google Scholar
25Jörgens, K. and Rellich, F.. Eigenwerttheorie gewöhnlicher Differentialgleichungen (Berlin: Springer, 1976).Google Scholar
26Kac, I. S.. On Hilbert spaces generated by monotonic Hermitian matrix functions. Zap. Nauchnoissled. Inst. Mat. Kharkov. Mat. Obshch. 22 (1950), 95–144 (in Russian).Google Scholar
27Kac, I. S.. On the multiplicity of the spectrum of a second order differential operator and the associated eigenfunction expansion. Izv. Akad. Nauk. SSSR, Ser. Mat. 27 (1963), 1081–112 (in Russian). Some of the results of this paper are announced in: I. S. Kac. On the multiplicity of the spectrum of a second order differential operator. Soviet Math. Dokl. 3 (1962), 1035–9 and some minor corrections are noted in: Izv. Akad. Nauk SSSR, Ser. Mat. 28 (1964), 951–2 (in Russian).Google Scholar
28Kato, T.. On finite dimensional perturbations of self adjoint operators. J. Math. Soc. Japan 9 (1957), 239–49.Google Scholar
29Kodaira, K.. The eigenvalue problem for ordinary differential equations of the second order and Heisenberg's theory of S-matrices. Amer. J. Math. 71 (1949), 921–45.Google Scholar
30Loomis, L. H.. The converse of the Fatou theorem for positive harmonic functions. Trans. Amer. Math. Soc. 53 (1943), 239–50.Google Scholar
31Naimark, M. A.. Linear Differential Operators, Part II (London: Harrap, 1968; translated from Russian).Google Scholar
32Pearson, D. B.. Singular continuous measures in scattering theory. Comm. Math. Phys. 60 (1978), 13–36.Google Scholar
33Pearson, D. B.. Quantum Scattering and Spectral Theory (London: Academic Press, 1988).Google Scholar
34Plesner, A. I.. Spectral Theory of Linear Operators, Vol. II (New York: F. Ungar, 1969; translated from Russian).Google Scholar
35Rofe-Beketov, F. S.. A test for the finiteness of the number of discrete levels introduced into the gaps of a continuous spectrum by perturbations of a periodic potential. Soviet Math. Dokl. 5 (1964), 689–92.Google Scholar
36Rosenblum, M.. Perturbation of the continuous spectrum and unitary equivalence. Pacific J. Math. 7 (1957), 997–1010.Google Scholar
37Saks, S.. Theory of the Integral, 2nd edn (New York: Hafner, 1937; translated from Polish).Google Scholar
38Stolz, G.. Bounded solutions and absolute continuity of Sturm–Liouville operators. J. Math. Anal. Appl. 169 (1992), 201–28.Google Scholar
39Stolz, G.. Spectral theory for slowly oscillating potentials II, Schrodinger operators. Math. Nachrichten 183 (1997), 275–94.Google Scholar
40Stone, M. H.. Linear Transformations on Hilbert Space, American Mathematical Society Colloquium Publications 15 (New York: American Mathematical Society, 1932).Google Scholar
41Strauss, A. V.. On the multiplicity of the spectrum of a self adjoint ordinary differential operator. Soviet Math. Dokl. 5 (1964), 524–7.Google Scholar
42Titchmarsh, E. C.. Eigenfunction Expansions associated with Second Order Differential Equations, Vol. I, 2nd edn (Oxford: Clarendon Press, 1962).Google Scholar
43Weidmann, J.. Linear Operators in Hilbert Space, Graduate Texts in Mathematics 68 (New York: Springer, 1980).Google Scholar
44Weidmann, J.. Spectral Theory of Ordinary Differential Operators, Lecture Notes in Mathematics 1258 (Heidelberg: Springer, 1987).Google Scholar
45Weyl, H.. Über gewöhnliche Differentialgleichungen mit singulären Stellen und ihre Eigenfunktionen. Nachr. Akad. Wiss. Göttingen, Math.-Phys. Kl. 1910 (1910), 442–67.Google Scholar
46Wolf, F.. Perturbation by changes in one-dimensional boundary conditions. Indag. Math. 18 (1956), 360–6.Google Scholar
47Wolfson, K. G.. On the spectrum of a boundary value problem with two singular endpoints. Amer. J. Math. 72 (1950), 713–9.Google Scholar