Hostname: page-component-586b7cd67f-dlnhk Total loading time: 0 Render date: 2024-11-27T14:34:54.652Z Has data issue: false hasContentIssue false
  • English
  • Français

On the Titchmarsh-Weyl M(λ)-coefficient and spectral density for a Dirac system

Published online by Cambridge University Press:  14 November 2011

Dominic P. Clemence
Dominic P. Clemence
Affiliation:
Department of Mathematics, Virginia Polytechnic Instituteand State University, Blacksburg, Virginia 24061, U.S.A.
Get access Rights & Permissions [Opens in a new window]

Synopsis

For a Dirac system on the line, we study the Jost solutions and the existence of half-bound states. A characterisation of this phenomenon is given in terms of the M(λ)-function, which supplements a well-known spectral characterisation for the system. As a corollary, we deduce the spectral density behaviour at the spectral gap endpoints, which is shown to depend on the possible existence of half-bound states.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 114 , Issue 3-4 , 1990 , pp. 259 - 277
Copyright
Copyright © Royal Society of Edinburgh 1990

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

1Coddington, E. A. and Levinson, N.. Theory of Ordinary Differential Equations (Malabar, Fl.: R. E. Krieger Publishing Co., 1984).Google Scholar
2Coppel, W. A.. Stability and Asymptotic Behaviour of Differential Equations (Boston: D. C. Heath and Co., 1965).Google Scholar
3Evgrafov, M. A.. Analytic Functions (Philadelphia: W. B. Saunders Co., 1966).Google Scholar
4Hinton, D. B. and Shaw, J. K.. Titchmarsh–Weyl theory for Hamiltonian systems. In Spectral Theory of Differential Operators, eds. Knowlesand, I. W. and Lewis, R. T., pp. 219–231.(New York: North Holland, 1981).Google Scholar
5Hinton, D. B. and Shaw, J. K.. On Titchmarsh–Weyl M(λ)–functions for linear Hamiltonian systems. J. Differential Equations 40 (1981), 316–342.CrossRefGoogle Scholar
6Hinton, D. B. and Shaw, J. K.. On the spectrum of a singular Hamiltonian system. Questiones Math. 5 (1982), 29–81.CrossRefGoogle Scholar
7Hinton, D. B. and Shaw, J. K.. Hamiltonian systems of limit point or limit cycle type with both endpoints singular. J. Differential Equations 50 (1983), 444–464.CrossRefGoogle Scholar
8Hinton, D. B. and Shaw, J. K.. On boundary value problems for Hamiltonian systems with two singular points. SIAM J. Math. Anal. 15 (1984), 272–286.CrossRefGoogle Scholar
9Hinton, D. B. and Shaw, J. K.. Absolutely continuous spectra of Dirac systems with long range, short range and oscillating potentials. Quart. J. Math. Oxford (2) 36 (1985), 183–213.CrossRefGoogle Scholar
10Hinton, D. B. and Shaw, J. K.. On the spectrum of a singular Hamiltonian system, II. Quaestiones Math. 10 (1986), 1–48.CrossRefGoogle Scholar
11Hinton, D. B. and Shaw, J. K.. Dirac systems with discrete spectra. Canad. J. Math. 49 (1987), 100–122.CrossRefGoogle Scholar
12Hinton, D. B., Klaus, M. and Shaw, J. K.. Levinson's theorem and Titchmarch–Weyl theory for Dirac systems. Proc. Roy. Soc. Edinburgh Sect A 109 (1988), 173–186.CrossRefGoogle Scholar
13Hinton, D. B., Mingarelli, A. B., Read, T. T. and Shaw, J. K.. On the numberof eigenvalues in the spectral gap of a Dirac system. Proc. Eninburgh Math. Soc. 29 (1986), 367–378.CrossRefGoogle Scholar
14Klaus, M.. On the point spectrum of Dirac operators. Helv. Phys. Ada 53 (1980), 453–462.Google Scholar
15Klaus, M. and Wüst, R.. Spectral properties of Dirac operators withsingular potentials. J. Math. Anal. Appl. 72 (1979), 206–214.CrossRefGoogle Scholar
16Klaus, M.. On the variation–diminishing property of Schrödinger operators. In CMS Conf. Proc.Vol. 8, pp. 199ö204. (Providence, RI: American Mathematical Society, 1986).Google Scholar
17Klaus, M.. Low energy behaviour of the scattering matrix for the Schrodinger equation on the line. Inverse Problems 4 (1988), 505–512.CrossRefGoogle Scholar
18Klaus, M.. Exact behaviour of Jost functions at low energy. J. Math. Phys. 29 (1988), 148–154.CrossRefGoogle Scholar
19Levitan, B. M. and Sargsjan, I.. Introduction to Spectral Theory: Self adjoint Ordinary Differential Operators. Translations of Mathematical Monographs 39 (Providence, RI: American Mathematical Society, 1975).CrossRefGoogle Scholar
20Newton, R. G.. Low energy scattering for medium range potentials. J. Math. Phys. 27 (1986), 2720–2730.CrossRefGoogle Scholar
21Barthélémy, M. C.. Contribution á l'étude de la diffusion par un potential central dans la théorie de l'electron de Dirac II. Ann. Inst. H. Poincaré A7 (1967), 115–143.Google Scholar
22Ablowitz, M. J. and Segur, H.. Solitons and the Inverse Scattering Transform (Philadelphia: SIAM, 1981).CrossRefGoogle Scholar
23Zakharov, V. E. and Shabat, A. B.. Interaction between solitons in a stable medium. Soviet Phys. JETP 37 (1973), 823–828.Google Scholar
24Zakharov, V. E. and Shabat, A. B.. Exact theory of two–dimensional self–focusing and one–dimensional self–modulation of waves in non–linear media. Soviet Phys. JETP 34 (1972), 62–69.Google Scholar