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Parameterization of the M(λ) function for a Hamiltonian system of limit circle type

  • D. B. Hinton (a1) and J. K. Shaw (a2)


The authors continue their study of Titchmarch-Weyl matrix M(λ) functions for linear Hamiltonian systems. A representation for the M(λ) function is obtained in the case where the system is limit circle, or maximum deficiency index, type. The representation reduces, in a special case, to a parameterization for scalar m-coefficients due to C. T. Fulton. A proof that matrix M(λ) functions are meromorphic in the limit circle case is given.



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1Atkinson, F. V.. Discrete and Continuous Boundary Problems (New York: Academic Press, 1964).
2Coddington, E. A. and Levinson, N.. Theory of Ordinary Differential Equations (New York: McGraw-Hill, 1955).
3Coppel, W. A.. Disconjugacy. Lecture Notes in Mathematics 220 (Berlin: Springer, 1971).
4Everitt, W. N. and Bennewitz, C.. Some remarks on the Titchmarsh-Weyl m-coefficient. In Tribute to Åke Pleijel, pp. 99108. Mathematics Department, University of Uppsala, Sweden, 1980.
5Fulton, C. T.. Parameterizations of Titchmarsh's m(λ)-functions inthe limit circle case. Trans. Amer. Math. Soc. 229 (1977), 5163.
6Hille, E.. Lectures on Ordinary Differential Equations (Reading, Mass.; Addison-Wesley, 1969).
7Hinton, D. B. and Shaw, J. K.. On Titchmarsh-Weyl m(λ)-functions for linear Hamiltonian systems. J. Differential Equations 40(3) (1981), 316342.
8Hinton, D. B. and Shaw, J. K.. On the spectrum of a singular Hamiltonian system. Quaestiones Math. 5 (1982), 2981
9Hinton, D. B. and Shaw, J. K.. Titchmarsh—Weyl theory for Hamiltonian systems. In Spectral Theory of Differential Operators, pp. 219321, Knowles, I. W. and Lewis, R. T. editors (New York: North-Holland, 1981).
10Kodaira, K.. The eigenvalue problem for ordinary differential equations of the second order and Heisenberg's theory of S-matrices. Amer. J. Math. 71 (1949), 921945.
11Kogan, V. I. and Rofe-Beketov, F. S., On square-integrable solutions of symmetric systems of differential equations of arbitrary order. Proc. Roy. Soc. Edinburgh Sect. A 74 (1974). 539.
12Krall, A. M.. Boundary values for an eigenvalue problem with a singular potential. J. Differential Equations 45 (1982). 128138.
13Stone, M. H.. Linear Transformations in Hilbert Space and their Applications to Analysis. Amer. Math. Soc. Colloq. Publ. vol. 15 (Providence, R.I.; Amer. Math. Soc, 1932).
14Titchmarsh, E. C.. Eigenfunction Expansions Associated with Second-Order Differential Equations, Pt I, 2nd edn (Oxford: Clarendon, 1962).
15Walker, P. W.. Adjoint boundary value problems for compactified singular differential operators, Pacific J. Math. 49 (1973), 265278.
16Weyl, H.. Über gewöhnliche differentialgleichungen mit singularitäten und die zugehörigen entwicklungen willkürlicher functionen. Math. Ann. 68 (1910), 220269.

Parameterization of the M(λ) function for a Hamiltonian system of limit circle type

  • D. B. Hinton (a1) and J. K. Shaw (a2)


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