Proceedings of the American Mathematical Society

Author(s): Gerald Teschl
Journal: Proc. Amer. Math. Soc. 126 (1998), 1685-1695.
MSC (1991): Primary 34C10, 39L40; Secondary 34B24, 34L15
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Abstract: Oscillation theory for one-dimensional Dirac operators with separated boundary conditions is investigated. Our main theorem reads: If $\lambda _{0,1}\in \mathbb R$ and if $u,v$

solve the Dirac equation $H u= \lambda _0 u$ , $H v= \lambda _1 v$ (in the weak sense) and respectively satisfy the boundary condition on the left/right, then the dimension of the spectral projection $P_{(\lambda _0, \lambda _1)}(H)$ equals the number of zeros of the Wronskian of $u$

and

. As an application we establish finiteness of the number of eigenvalues in essential spectral gaps of perturbed periodic Dirac operators.

1.: W. Bulla, F. Gesztesy, and K. Unterkofler On relativistic energy band corrections in the presence of periodic potentials, Lett. Math. Phys. 15, 313-324 (1988). MR 90c:81038
2.: W.A. Coppel Disconjugacy, Lecture Notes in Mathematics 220, Springer, Berlin 1971. MR 57:778
3.: I.S. Frolov, Inverse scattering problem for a Dirac system on the whole axis, Soviet Math. Dokl. 13, 1468-1472 (1972). MR 47:5352
4.: F. Gesztesy, B. Simon, and G. Teschl, Zeros of the Wronskian and renormalized oscillation theorems, Am. J. Math. 118, 571-594 (1996). CMP 96:13
5.: P. Hartman, Differential equations with non-oscillatory eigenfunctions, Duke Math. J. 15, 697-709 (1948). MR 10:376e
6.: P. Hartman, A characterization of the spectra of one-dimensional wave equations, Am. J. Math. 71, 915-920 (1949). MR 11:438a
7.: P. Hartman and C.R. Putnam, The least cluster point of the spectrum of boundary value problems, Am. J. Math. 70, 849-855 (1948). MR 10:376f
8.: D.B. Hinton and C.K. Shaw, Asymptotics of solutions and spectra of perturbed periodic Hamiltonian systems, in "Differential Equations and Mathematical Physics", (I.W. Knowles and Y. Saito Eds.), 169-174, Lecture Notes in Mathematics 1285, Springer, Berlin 1987. MR 89g:34026
9.: D.B. Hinton and C.K. Shaw, Absolutely continuous spectra of perturbed periodic Hamiltonian systems, Rocky Mtn. J. Math., 727-748 (1987). MR 89m:34031
10.: D.B. Hinton, A.B. Mingarelli, T.T. Read, and C.K. Shaw, On the number of eigenvalues in the spectral gap of a Dirac system, Proc. of the Edinburgh Math. Soc. 29, 367-378 (1986). MR 88d:34029
11.: K. Kreith Oscillation Theory, Lecture Notes in Mathematics 324, Springer, Berlin 1973.
12.: B.M. Levitan and I.S. Sargsjan, Sturm-Liouville and Dirac Operators, Kluwer Academic Publishers, Dordrecht 1991. MR 92i:34119
13.: M. Reed and B. Simon, Methods of Modern Mathematical Physics IV. Analysis of Operators, Academic Press, San Diego, 1978. MR 58:12429c
14.: W.T. Reid Sturmian Theory for Ordinary Differential Equations, Springer, New York 1980. MR 82f:34002
15.: F.S. Rofe-Beketov, A test for the finiteness of the number of discrete levels introduced into gaps of a continuous spectrum by perturbations of a periodic potential, Soviet Math. Dokl. 5, 689-692 (1964). MR 28:4176
16.: J.C.F. Sturm Mémoire sur les équations différentielles linéaires du second ordre, J. Math. Pures Appl. 1, 106-186 (1836).
17.: C.A. Swanson Comparison and Oscillation Theory of Linear Differential Equations, Academic Press, New York 1968. MR 57:3515
18.: G. Teschl, Oscillation theory and renormalized oscillation theory for Jacobi operators, J. Diff. Eqs. 129, 532-558 (1996). CMP 96:17
19.: B. Thaller, The Dirac Equation, Springer, Berlin, 1992. MR 94k:81056
20.: S. Timischl, A trace formula for one-dimensional Dirac operators, diploma thesis, University of Graz, Austria, 1995.
21.: K. Unterkofler, Periodische Potentiale in der eindimensionalen Diracgleichung, diploma thesis, Technical University of Graz, Austria, 1986.
22.: J. Weidmann, Spectral Theory of Ordinary Differential Operators, Lecture Notes in Mathematics 1258, Springer, Berlin 1987. MR 89b:47070
23.: J. Weidmann, Zur Spektraltheorie von Sturm-Liouville Operatoren, Math. Z. 98, 268-302 (1971). MR 35:4769
24.: J. Weidmann, Ozillationsmethoden für Systeme gewönlicher Differentialgleichungen, Math. Z. 119, 349-337 (1971). MR 44:2975
25.: J. Weidmann, Absolut stetiges Spektrum bei Sturm-Liouville und Dirac-Systemen, Math. Z. 180, 423-427 (1982). MR 83m:34023

Retrieve articles in Proceedings of the American Mathematical Society with MSC (1991): 34C10, 39L40, 34B24, 34L15

Gerald Teschl
Affiliation: Institut für Reine und Angewandte Mathematik RWTH Aachen 52056 Aachen Germany
Address at time of publication: Institut für Mathematik, Universität Wien, Strudelhofgasse 4, 1090 Vienna, Austria
Email: gerald@mat.univie.ac.at

DOI: 10.1090/S0002-9939-98-04310-X
PII: S 0002-9939(98)04310-X
Keywords: Oscillation theory, Dirac operators, spectral theory
Received by editor(s): November 7, 1996
Communicated by: Palle E. T. Jorgensen
Copyright of article: Copyright 1998, by the author

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