Strong limit-point classification of singular Hamiltonian expressions

Qi, Jiangang; Chen, Shaozhu

doi:10.1090/S0002-9939-04-07037-6

Strong limit-point classification of singular Hamiltonian expressions
HTML articles powered by AMS MathViewer

by Jiangang Qi and Shaozhu Chen PDF

Proc. Amer. Math. Soc. 132 (2004), 1667-1674 Request permission

Abstract:

Strong limit-point criteria for singular Hamiltonian differential expressions with complex coefficients are obtained. The main results are extensions of the previous results due to Everitt, Giertz, and Weidmann for scalar differential expressions.

References

F. V. Atkinson, Discrete and continuous boundary problems, Mathematics in Science and Engineering, Vol. 8, Academic Press, New York-London, 1964. MR 0176141
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. MR 1069450, DOI 10.1016/0022-247X(90)90245-B
Nelson Dunford and Jacob T. Schwartz, Linear operators. Part II: Spectral theory. Self adjoint operators in Hilbert space, Interscience Publishers John Wiley & Sons, New York-London, 1963. With the assistance of William G. Bade and Robert G. Bartle. MR 0188745
W. N. Everitt, On the limit-point classification of second-order differential operators, J. London Math. Soc. 41 (1966), 531–534. MR 200519, DOI 10.1112/jlms/s1-41.1.531
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. MR 303355, DOI 10.1112/plms/s3-24.4.756
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. MR 361255, DOI 10.1112/plms/s3-29.1.142
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. MR 326047, DOI 10.1007/BF01428264
W. N. Everitt, D. B. Hinton, and J. S. W. Wong, On the strong limit-$n$ classification of linear ordinary differential expressions of order $2n$, Proc. London Math. Soc. (3) 29 (1974), 351–367. MR 409956, DOI 10.1112/plms/s3-29.2.351
Einar Hille, Lectures on ordinary differential equations, Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont., 1969. MR 0249698
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. MR 731867, DOI 10.1137/0515022
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. MR 790479, DOI 10.1093/qmath/36.2.183
Roger A. Horn and Charles R. Johnson, Matrix analysis, Cambridge University Press, Cambridge, 1985. MR 832183, DOI 10.1017/CBO9780511810817
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. MR 0481243
Allan M. Krall, $M(\lambda )$ theory for singular Hamiltonian systems with one singular point, SIAM J. Math. Anal. 20 (1989), no. 3, 664–700. MR 990871, DOI 10.1137/0520047
Allan M. Krall, A limit-point criterion for linear Hamiltonian systems, Appl. Anal. 61 (1996), no. 1-2, 115–119. MR 1625457, DOI 10.1080/00036819608840449
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.
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. MR 369792, DOI 10.1112/jlms/s2-9.1.151