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Asymptotics of Sturm-Liouville eigenvalues for problems on a finite interval with one limit-circle singularity, I*

Published online by Cambridge University Press:  14 November 2011

F. V. Atkinson
Affiliation:
Department of Mathematics, University of Toronto, Toronto, Canada
C. T. Fulton
Affiliation:
Department of Mathematical Sciences, Florida Institute of Technology, Melbourne, FL 32901, U.S.A.

Extract

Asymptotic formulae for the positive eigenvalues of a limit-circle eigenvalue problem for –y” + qy = λy on the finite interval (0, b] are obtained for potentials q which are limit circle and non-oscillatory at x = 0, under the assumption xq(x)∈L1(0,6). Potentials of the form q(x) = C/xk, 0<fc<2, are included. In the case where k = 1, an independent check based on the limit-circle theory of Fulton and an asymptotic expansion of the confluent hypergeometric function, M(a, b; z), verifies the main result.

Type
Research Article
Copyright
Copyright © Royal Society of Edinburgh 1984

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References

1Abramowitz, M. and Stegun, I. A.. Handbook of mathematical functions (NBS Applied Math Series 55) (U.S. Department of Commerce, 1964).Google Scholar
2Atkinson, F. V. and Fulton, C. T.. Asymptotic formulae for eigenvalues of limit circle problems on a half line. Ann. Mat. Pura Appl. 85 (1983), 363398.CrossRefGoogle Scholar
3Borg, G.. Eine Umkehrung der Sturm-Liouvilleschen Eigenwertaufgabe. Ada Math. 78 (1946), 196.Google Scholar
4Burnap, C., Greenberg, W. and Zweifel, P. F.. Eigenvalue problem for singular potentials. Nuovo Cimento 50 (1979), 457465.CrossRefGoogle Scholar
5Calogero, F.. Approximation for the phase shifts produced by repulsive potentials strongly singular at the origin. Phys. Rev. 135B (1964), 693700.CrossRefGoogle Scholar
6Case, K. M.. Singular potentials. Phys. Rev. 80 (1950), 797806.CrossRefGoogle Scholar
7Dunford, N. and Schwartz, J. T.. Linear operators, II (New York: Interscience, 1963).Google Scholar
8Engelke, R. and Beckel, C. L.. Nonuniqueness of the energy correction in application of the WKB approximation to radial problems. J. Math. Phys. 11 (1970), 19911994.CrossRefGoogle Scholar
9Fulton, C. T., Paratnetrizations of Titchmarsh's m(λ)-functions in the limit circle case. Trans. Amer. Math. Soc. 229 (1977), 5163.Google Scholar
10Fulton, C. T., An integral equation iterative scheme for asymptotic expansions of spectral quantities of regular Sturm-Liouville problems. J. Integral Equations 4 (1982), 163172.Google Scholar
11Fulton, C. T.. Two point boundary value problems with eigenvalue parameter contained in the boundary conditions. Proc. Roy. Soc. Edinburgh Sect. A 77 (1977), 293308.CrossRefGoogle Scholar
12Hartman, P.. Ordinary differential equations (New York: Wiley, 1964).Google Scholar
13Hellwig, G.. Differential operators of mathematical physics (Engl. Transl., Reading, Mass.: Addison-Wesley, 1967).Google Scholar
14Hobson, E. W.. On a general convergence theorem, and the theory of the representation of a function by series of normal functions. Proc. London Math. Soc. 6 (1908), 349395.CrossRefGoogle Scholar
15Krall, A. M.. Boundary values for an eigenvalue problem with a singular potential. J. Differential Equations 45 (1982), 128138.CrossRefGoogle Scholar
16Levitan, B. M. and Gasymov, M. G.. Determination of a differential equation by two of its spectra. Russian Math. Surveys 19 (1964), 163.CrossRefGoogle Scholar
17Meetz, K.. Singular potentials in nonrelativistic quantum mechanics. Nuovo Cimento 34 (1964), 690708.CrossRefGoogle Scholar
18O'Malley, T. F., Spurch, L. and Rosenberg, L.. Modification of effective-range theory in the presence of a long range (r−4) potential. J. Math. Phys. 4 (1961), 491498.CrossRefGoogle Scholar
19Rellich, F.. Halbbeschrankte gewohnliche Differentialoperatoren zweiter Ordnung. Math. Ann. 122 (1951), 343368.CrossRefGoogle Scholar
20Sears, D. B.. Sturm-Liouville Theory I. Lecture Notes (Adelaide: Flinders Univ., 1971).Google Scholar
21Titchmarsh, E. C.. Eigenfunction expansions associated with second order differential equations, I (2nd edn) (Oxford: Clarendon, 1962).Google Scholar
22Ulehla, I., Havlicek, M. and Horejsi, J.. Eigenvalues of the Schrodinger operator via Priifer transformation. Phys. Lett. A 82 (1981), 6466.CrossRefGoogle Scholar
23Vogt, E. and Wannier, G. H.. Scattering of ions by polarization forces. Phys. Rev. 95 (1954), 11901198.CrossRefGoogle Scholar
24Wray, S. D.. Absolutely convergent Sturm-Liouville expansions. Proc. Roy. Soc. Edinburgh Sect A 73 (1974/1975), 254269.Google Scholar