Deforming the point spectra of one-dimensional Dirac operators

Teschl, Gerald

doi:10.1090/S0002-9939-98-04362-7

Deforming the point spectra of one-dimensional Dirac operators
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by Gerald Teschl PDF

Proc. Amer. Math. Soc. 126 (1998), 2873-2881

Abstract:

We provide a method of inserting and removing any finite number of prescribed eigenvalues into spectral gaps of a given one-dimensional Dirac operator. This is done in such a way that the original and deformed operators are unitarily equivalent when restricted to the complement of the subspace spanned by the newly inserted eigenvalue. Moreover, the unitary transformation operator which links the original operator to its deformed version is explicitly determined.

References

J. Barcelos-Neto, Ashok Das, and W. Scherer, Dirac quantization of the chiral superfield, Phys. Rev. D (3) 34 (1986), no. 4, 1108–1113. MR 852932, DOI 10.1103/PhysRevD.34.1108
P. A. Deift, Applications of a commutation formula, Duke Math. J. 45 (1978), no. 2, 267–310. MR 495676, DOI 10.1215/S0012-7094-78-04516-7
P. Deift and E. Trubowitz, Inverse scattering on the line, Comm. Pure Appl. Math. 32 (1979), no. 2, 121–251. MR 512420, DOI 10.1002/cpa.3160320202
F. R. Gantmacher, Matrizenrechnung. II. Spezielle Fragen und Anwendungen, Hochschulbücher für Mathematik, Band 37, VEB Deutscher Verlag der Wissenschaften, Berlin, 1959 (German). MR 0107647
Saunders MacLane, Steinitz field towers for modular fields, Trans. Amer. Math. Soc. 46 (1939), 23–45. MR 17, DOI 10.1090/S0002-9947-1939-0000017-3
F. Gesztesy, A complete spectral characterization of the double commutation method, J. Funct. Anal. 117 (1993), no. 2, 401–446. MR 1244942, DOI 10.1006/jfan.1993.1132
Fritz Gesztesy and Roman Svirsky, (m)KdV solitons on the background of quasi-periodic finite-gap solutions, Mem. Amer. Math. Soc. 118 (1995), no. 563, iv+88. MR 1303091, DOI 10.1090/memo/0563
F. Gesztesy and G. Teschl, On the double commutation method, Proc. Amer. Math. Soc. 124 (1996), no. 6, 1831–1840. MR 1322925, DOI 10.1090/S0002-9939-96-03299-6
F. Gesztesy, B. Simon, and G. Teschl, Spectral deformations of one-dimensional Schrödinger operators, J. d’Analyse Math. 70, (1996), 267–324.
B. M. Levitan and I. S. Sargsjan, Sturm-Liouville and Dirac operators, Mathematics and its Applications (Soviet Series), vol. 59, Kluwer Academic Publishers Group, Dordrecht, 1991. Translated from the Russian. MR 1136037, DOI 10.1007/978-94-011-3748-5
Rafael R. del Río Castillo, Embedded eigenvalues of Sturm-Liouville operators, Comm. Math. Phys. 142 (1991), no. 2, 421–431. MR 1137069, DOI 10.1007/BF02102068
Bernd Thaller, The Dirac equation, Texts and Monographs in Physics, Springer-Verlag, Berlin, 1992. MR 1219537, DOI 10.1007/978-3-662-02753-0
Joachim Weidmann, Spectral theory of ordinary differential operators, Lecture Notes in Mathematics, vol. 1258, Springer-Verlag, Berlin, 1987. MR 923320, DOI 10.1007/BFb0077960
Joachim Weidmann, Oszillationsmethoden für Systeme gewöhnlicher Differentialgleichungen, Math. Z. 119 (1971), 349–373 (German). MR 285758, DOI 10.1007/BF01109887