Abstract:
A trace formula is given for an abstract pair consisting of a dissipative operator and a selfadjoint operator, and a connection is established between the spectral shift function of this pair and the corresponding scattering matrix. As a consequence, trace formulas are obtained for a specific dissipative operator arising in the problem of resonance scattering of plane waves on a one-dimensional semi-infinite crystal.
Bibliography: 8 titles.
Citation:
A. V. Rybkin, “The spectral shift function for a dissipative and a selfadjoint operator, and trace formulas for resonances”, Math. USSR-Sb., 53:2 (1986), 421–431
\Bibitem{Ryb84}
\by A.~V.~Rybkin
\paper The spectral shift function for a dissipative and a selfadjoint operator, and trace formulas for resonances
\jour Math. USSR-Sb.
\yr 1986
\vol 53
\issue 2
\pages 421--431
\mathnet{http://mi.mathnet.ru/eng/sm2093}
\crossref{https://doi.org/10.1070/SM1986v053n02ABEH002930}
\mathscinet{http://mathscinet.ams.org/mathscinet-getitem?mr=764904}
\zmath{https://zbmath.org/?q=an:0586.47034|0569.47033}
Linking options:
https://www.mathnet.ru/eng/sm2093
https://doi.org/10.1070/SM1986v053n02ABEH002930
https://www.mathnet.ru/eng/sm/v167/i3/p420
This publication is cited in the following 12 articles:
V. V. Peller, “Besov spaces in operator theory”, Russian Math. Surveys, 79:1 (2024), 1–52
M. M. Malamud, H. Neidhardt, V. V. Peller, “Real-Valued Spectral Shift Functions for Contractions and Dissipative Operators”, Dokl. Math., 2024
M. M. Malamud, H. Neidhardt, V. V. Peller, “Analytic operator Lipschitz functions in the disk and a trace formula for functions of contractions”, Funct. Anal. Appl., 51:3 (2017), 185–203
Behrndt J., Malamud M.M., Neidhardt H., “Trace Formulae for Dissipative and Coupled Scattering Systems”, Spectral Theory in Inner Product Spaces and Applications, Operator Theory Advances and Applications, 188, eds. Behrndt J., Forster KH., Langer H., Trunk C., Birkhauser Verlag Ag, 2009, 49–85
A. V. Rybkin, “The spectral shift function, the characteristic function of a contraction, and a generalized integral”, Russian Acad. Sci. Sb. Math., 83:1 (1995), 237–281
V. A. Lyubishkin, “On the trace formulas of Gel'fand–Levitan and Krein”, Math. USSR-Sb., 74:2 (1993), 531–540
Adamjan V. Neidhardt H., “On the Summability of the Spectral Shift Function for Pair of Contractions and Dissipative Operators”, 24, no. 1, 1990, 187–206
A. V. Rybkin, “The discrete and the singular spectrum in the trace formula for contracting and unitary operators”, Funct. Anal. Appl., 23:3 (1989), 244–246
Neidhardt H., “Scattering Matrix and Spectral Shift of the Nuclear Dissipative Scattering-Theory .2.”, 19, no. 1, 1988, 43–62
Jonas P., “On the Trace Formula of Perturbation-Theory .1.”, Math. Nachr., 137 (1988), 257–281
A. V. Rybkin, “Trace for the pair of a contraction and a unitary operator”, Funct. Anal. Appl., 21:4 (1987), 334–336
Krein M., “Disturbance Determinants and the Trace Formula for Some Classes of Operator Pairs”, 17, no. 1, 1987, 129–187
Citation in format AMSBIB
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