×

Strong limit-point classification of singular Hamiltonian expressions. (English) Zbl 1054.34041

The authors investigate singular Hamiltonian differential systems with complex coefficients of the form \(Lz:=Jz'-Q(t)z=\lambda P(t)z\), where \(\lambda\) is a complex parameter, \[ J=\left(\begin{matrix} 0&-I\\ I&0 \end{matrix}\right),\quad Q(t)=\left( \begin{matrix} -C(t)&A^\ast(t)\\ A(t)&B(t) \end{matrix}\right),\quad P(t)=\left( \begin{matrix} W(t)& 0\\ 0&0 \end{matrix}\right), \] \(A,\;B,\;C,\;W\) are locally integrable \(n\times n\)-matrix-valued functions on \([0,\infty)\), \(B,C,W\) are Hermitian with \(W>0\). They give strong limit-point criteria for the operator \(L\) generated by the system, which extend the results due to Everitt, Giertz and Weidmann for the scalar case, see [W. N. Everitt, J. Lond. Math. Soc. 41, 531–534 (1966; Zbl 0145.10604) and W. N. Everitt, M. Giertz and J. Weidmann, Math. Ann. 200, 335–346 (1973; Zbl 0235.34045)].
Reviewer: Pavel Rehak (Brno)

MSC:

34B20 Weyl theory and its generalizations for ordinary differential equations
37J99 Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] F. V. Atkinson, Discrete and continuous boundary problems, Mathematics in Science and Engineering, Vol. 8, Academic Press, New York-London, 1964. · Zbl 0117.05806
[2] Stephen L. Clark, A criterion for absolute continuity of the continuous spectrum of a Hamiltonian system, J. Math. Anal. Appl. 151 (1990), no. 1, 108 – 128. · Zbl 0735.34067
[3] Nelson Dunford and Jacob T. Schwartz, Linear operators. Part II: Spectral theory. Self adjoint operators in Hilbert space, With the assistance of William G. Bade and Robert G. Bartle, Interscience Publishers John Wiley & Sons New York-London, 1963. · Zbl 0128.34803
[4] W. N. Everitt, On the limit-point classification of second-order differential operators, J. London Math. Soc. 41 (1966), 531 – 534. · Zbl 0145.10604
[5] W. N. Everitt and M. Giertz, On some properties of the domains of powers of certain differential operators, Proc. London Math. Soc. (3) 24 (1972), 756 – 768. · Zbl 0258.34007
[6] W. N. Everitt, M. Giertz, and J. B. McLeod, On the strong and weak limit-point classfication of second-order differential expressions, Proc. London Math. Soc. (3) 29 (1974), 142 – 158. · Zbl 0302.34022
[7] W. N. Everitt, M. Giertz, and J. Weidmann, Some remarks on a separation and limit-point criterion of second-order, ordinary differential expressions, Math. Ann. 200 (1973), 335 – 346. · Zbl 0235.34045
[8] W. N. Everitt, D. B. Hinton, and J. S. W. Wong, On the strong limit-\? classification of linear ordinary differential expressions of order 2\?, Proc. London Math. Soc. (3) 29 (1974), 351 – 367. · Zbl 0305.34040
[9] Einar Hille, Lectures on ordinary differential equations, Addison-Wesley Publ. Co., Reading, Mass.-London-Don Mills, Ont., 1969. · Zbl 0179.40301
[10] D. B. Hinton and J. K. Shaw, On boundary value problems for Hamiltonian systems with two singular points, SIAM J. Math. Anal. 15 (1984), no. 2, 272 – 286. · Zbl 0539.34016
[11] D. B. Hinton and J. K. Shaw, Absolutely continuous spectra of Dirac systems with long range, short range and oscillating potentials, Quart. J. Math. Oxford Ser. (2) 36 (1985), no. 142, 183 – 213. · Zbl 0571.34017
[12] Roger A. Horn and Charles R. Johnson, Matrix analysis, Cambridge University Press, Cambridge, 1985. · Zbl 0576.15001
[13] Robert M. Kauffman, Thomas T. Read, and Anton Zettl, The deficiency index problem for powers of ordinary differential expressions, Lecture Notes in Mathematics, Vol. 621, Springer-Verlag, Berlin-New York, 1977. · Zbl 0367.34014
[14] Allan M. Krall, \?(\?) theory for singular Hamiltonian systems with one singular point, SIAM J. Math. Anal. 20 (1989), no. 3, 664 – 700. · Zbl 0683.34008
[15] Allan M. Krall, A limit-point criterion for linear Hamiltonian systems, Appl. Anal. 61 (1996), no. 1-2, 115 – 119. · Zbl 0876.34027
[16] V. K. Kumar, On the strong limit-point classification of fourth-order differential expressions with complex coefficients, J. London Math. Soc. (2) 12 (1976), 287-298. · Zbl 0325.34021
[17] Philip W. Walker, A vector-matrix formulation for formally symmetric ordinary differential equations with applications to solutions of integrable square, J. London Math. Soc. (2) 9 (1974/75), 151 – 159. · Zbl 0308.34011
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.