@article{JMAG_2008_4_1_a3,
author = {F. Gesztesy and A. Pushnitski and B. Simon},
title = {On the {Koplienko} spectral shift function. {I.~Basics}},
journal = {\v{Z}urnal matemati\v{c}eskoj fiziki, analiza, geometrii},
pages = {63--107},
year = {2008},
volume = {4},
number = {1},
language = {en},
url = {http://geodesic.mathdoc.fr/item/JMAG_2008_4_1_a3/}
}
TY - JOUR AU - F. Gesztesy AU - A. Pushnitski AU - B. Simon TI - On the Koplienko spectral shift function. I. Basics JO - Žurnal matematičeskoj fiziki, analiza, geometrii PY - 2008 SP - 63 EP - 107 VL - 4 IS - 1 UR - http://geodesic.mathdoc.fr/item/JMAG_2008_4_1_a3/ LA - en ID - JMAG_2008_4_1_a3 ER -
F. Gesztesy; A. Pushnitski; B. Simon. On the Koplienko spectral shift function. I. Basics. Žurnal matematičeskoj fiziki, analiza, geometrii, Tome 4 (2008) no. 1, pp. 63-107. http://geodesic.mathdoc.fr/item/JMAG_2008_4_1_a3/
[1] V. M. Adamjan, H. Neidhardt, “On the Summability of the Spectral Shift Function for Pair of Contractions and Dissipative Operators”, J. Oper. Th., 24 (1990), 187–206 | MR | Zbl
[2] Vestnik Leningrad. Univ. Mat. Mekh. Astronom., 118:2 (1979), 5–9 | MR
[3] R. Alicki, M. Fannes, Quantum Dynamical Systems, Oxford Univ. Press, Oxford, 2001 | MR | Zbl
[4] N. Aronszajn, “On a Problem of Weyl in the Theory of Singular Sturm–Liouville Equations”, Amer. J. Math., 79 (1957), 597–610 | DOI | MR | Zbl
[5] N. Aronszajn, W. F. Donoghue, “On Exponential Representations of Analytic Functions in the Upper Half-plane with Positive Imaginary Part”, Amer. J. Math., 5 (1956–1957), 321–388
[6] J. Avron, R. Seiler, B. Simon, “The Index of a Pair of Projections”, J. Funct. Anal., 120 (1994), 220–237 | DOI | MR | Zbl
[7] H. Baumgärtel, M. Wollenberg, Mathematical Scattering Theory, Operator Theory: Adv. Appl., 9, Birkhäuser, Boston, 1983 | DOI | MR | Zbl
[8] J. Bendat, S. Sherman, “Monotone and Convex Operator Functions”, Trans. Amer. Math. Soc., 79 (1955), 58–71 | DOI | MR | Zbl
[9] R. Bhatia, Matrix Analysis, Springer, New York, 1997 | MR
[10] Dokl. Akad. Nauk USSR, 147:5 (1962), 1008–1009 | MR | Zbl
[11] M. Sh. Birman, “Existence Conditions for Wave Operators”, Izv. Akad. Nauk USSR. Ser. Mat., 27 (1963), 883–906 (Russian) | MR | Zbl
[12] Dokl. Akad. Nauk USSR, 165 (1965), 1223–1226 | MR | Zbl
[13] Probl. Math. Phys., v. 1, Spectr. Th. and Wave Processes, Izd-vo Leningr. Univ., 1966, 33–67 | MR
[14] “Boundary Value Problems of Mathematical Physics and Related Questions in the Theory of Functions, 6”, Zap. Naučn. Sem. Leningrad. Otdel. Steklov Mat. Inst. (LOMI), 27, 1972, 33–46 | DOI | MR | MR | Zbl | Zbl
[15] M. Sh. Birman, M. Solomyak, “On Double Operator Integrals in a Hilbert Space”, Integr. Eq. Oper. Th., 47 (2003), 131–168 | DOI | MR | Zbl
[16] M. Sh. Birman, D. R. Yafaev, “The Spectral Shift Function”, The Work of M. G. Krein and its Further Development, St. Petersburg Math. J., 4 (1993), 833–870 | MR
[17] M. Christ, A. Kiselev, “Absolutely Continuous Spectrum for One-Dimensional Schrödinger Operators with Slowly Decaying Potentials: Some Optimal Results”, J. Amer. Math. Soc., 11 (1998), 771–797 | DOI | MR | Zbl
[18] P. A. Deift, R. Killip, “On the Absolutely Continuous Spectrum of One-Dimensional Schrödinger Operators with Square Summable Potentials”, Comm. Math. Phys., 203 (1999), 341–347 | DOI | MR | Zbl
[19] M. Dostanić, “Trace Formula for Nonnuclear Perturbations of Selfadjoint Operators”, Publ. Inst. Mathématique, 54(68) (1993), 71–79 | MR | Zbl
[20] N. Dunford, J. T. Schwartz, Linear Operators. Part II. Spectral Theory, Interscience, New York, 1998 | MR
[21] P. L. Duren, “On the Bloch–Nevanlinna Conjecture”, Colloq. Math., 20 (1969), 295–297 | MR | Zbl
[22] P. L. Duren, Theory of $H^p$ Spaces, Acad. Press, Boston, 1970 | MR
[23] E. G. Effros, “Why the Circle Is Connected: An Introduction to Quantized Topology”, Math. Intelligencer, 11 (1989), 27–34 | DOI | MR | Zbl
[24] F. Gesztesy, K. A. Makarov, A. K. Motovilov, “Monotonicity and Concavity Properties of the Spectral Shift Function”, Stochastic Processes, Physics and Geometry: New Interplays, v. II, Proc. Canad. Math. Soc. Conf., 29, A Volume in Honor of Sergio Albeverio, eds. F. Gesztesy, H. Holden, J. Jost, S. Paycha, M. Röckner, S. Scarlatti, AMS, Providence, RI, 2000, 207–222 | MR | Zbl
[25] F. Gesztesy, A. Pushnitski, B. Simon, preprint, in preparation, in press
[26] F. Gesztesy, B. Simon, “Rank One Perturbations at Infinite Coupling”, J. Funct. Anal., 128 (1995), 245–252 | DOI | MR | Zbl
[27] I. C. Gohberg, M. G. Krein, Introduction to the Theory of Linear Nonselfadjoint Operators, Transl. Math. Monographs, 18, AMS, Providence, RI, 1969 | MR | Zbl
[28] R. B. Israel, Convexity in the Theory of Lattice Gases, Princeton Univ. Press, Princeton, 1979 | MR | Zbl
[29] P. Jonas, “On the Trace Formula for Perturbation Theory, I”, Math. Nachr., 137 (1988), 257–281 | DOI | MR | Zbl
[30] P. Jonas, “On the Trace Formula for Perturbation Theory, II”, Math. Nachr., 197 (1999), 29–49 | DOI | MR | Zbl
[31] T. Kato, “On Finite-Dimensional Perturbations of Selfadjoint Operators”, Math. Soc. Japan, 9 (1957), 239–249 | DOI | MR | Zbl
[32] T. Kato, “Perturbation of Continuous Spectra by Trace Class Operators”, Proc. Japan Acad., 33 (1957), 260–264 | DOI | MR | Zbl
[33] T. Kato, Perturbation Theory for Linear Operators, Corr. Printing of the 2nd Edition, Grundlehren der Mathematischen Wissenschaften, 132, Springer, Berlin, 1980 | MR | Zbl
[34] R. Killip, B. Simon, “Sum Rules for Jacobi Matrices and their Applications to Spectral Theory”, Ann. Math. (2), 158 (2003), 253–321 | DOI | MR | Zbl
[35] O. Klein, “Zur Quantenmechanischen Begründung des Zweiten Haupsatzes der Wärmelehre”, Z. Phys., 72 (1931), 767–775 | DOI | Zbl
[36] Sib. Mat. Zh., 25 (1984), 62–71 | DOI | MR | Zbl
[37] F. Kraus, “Über konvexe Matrixfunktionen”, Math. Z., 41 (1936), 18–42 | DOI | MR
[38] M. G. Krein, “On the Trace Formula in Perturbation Theory”, Mat. Sb., 33(75) (1953), 597–626 (Russian) | MR
[39] Dokl. Akad. Nauk USSR, 144 (1962), 268–271 | MR
[40] M. G. Krein, “On Certain New Studies in the Perturbation Theory for Selfadjoint Operators”, Topics in Differential and Integral Equations and Operator Theory, Operator Theory: Adv. Appl., 7, ed. I. Gohberg, Birkhäuser, Basel, 1983, 107–172 | DOI | MR
[41] M. G. Krein, “On Perturbation Determinants and a Trace Formula for Certain Classes of Pairs of Operators”, Amer. Math. Soc. Transl. (2), 145 (1989), 39–84 | Zbl
[42] M. G. Krein, V. A. Yavryan, “Spectral Shift Functions that Arise in Perturbations of a Positive Operator”, J. Oper. Th., 6 (1981), 155–191 (Russian) | MR
[43] S. T. Kuroda, “On a Theorem of Weyl–von Neumann”, Proc. Japan Acad., 34 (1958), 11–15 | DOI | MR | Zbl
[44] H. Langer, “Eine Erweiterung der Spurformel der Störungstheorie”, Math. Nachr., 30 (1965), 123–135 | DOI | MR | Zbl
[45] E. Lieb, G. K. Pedersen, “Convex Multivariable Trace Functions”, Rev. Math. Phys., 14 (2002), 631–648 | DOI | MR | Zbl
[46] K. Löwner, “Über Monotone Matrixfunktionen”, Math. Z., 38 (1934), 177–216 | DOI | MR
[47] H. Neidhardt, “Scattering Matrix and Spectral Shift of the Nuclear Dissipative Scattering Theory”, Operators in Indefinite Metric Spaces, Scattering Theory and Other Topics, Operator Theory: Adv. Appl., 24, eds. H. Helson, G. Arsene, Birkhäuser, Basel, 1987, 237–250 | MR
[48] H. Neidhardt, “Scattering Matrix and Spectral Shift of the Nuclear Dissipative Scattering Theory, II”, J. Oper. Th., 19 (1988), 43–62 | MR | Zbl
[49] H. Neidhardt, “Spectral Shift Function and Hilbert–Schmidt Perturbation: Extensions of Some Work of Koplienko”, Math. Nachr., 138 (1988), 7–25 | DOI | MR | Zbl
[50] Funkts. Anal. Prilozh., 19 (1985), 37–51 | DOI | MR | Zbl
[51] V. V. Peller, “Hankel Operators in the Perturbation Theory of Unbounded Selfadjoint Operators”, Anal. Part. Diff. Eq., Lecture Notes in Pure and Appl. Math., 122, ed. C. Sadosky, Dekker, New York, 1990, 529–544 | MR
[52] V. V. Peller, “An Extension of the Koplienko–Neidhardt Trace Formulae”, J. Funct. Anal., 221 (2005), 456–481 | DOI | MR | Zbl
[53] M. Reed, B. Simon, Methods of Modern Mathematical Physics. IV. Analysis of Operators, Acad. Press, New York, 1978 | MR | Zbl
[54] M. Rosenblum, “Perturbation of the Continuous Spectrum and Unitary Equivalence”, Pacific J. Math., 7 (1957), 997–1010 | DOI | MR | Zbl
[55] D. Ruelle, Statistical Mechanics. Rigorous Results, Reprint of the 1989 Edition, World Sci. Publ., River Edge, NJ. ; Imperial College Press, London, 1999 | MR
[56] Mat. Sb., 185:10 (1994), 91–144 | DOI | MR | Zbl
[57] A. V. Rybkin, “On $A$-integrability of the Spectral Shift Function of Unitary Operators Arising in the Lax–Phillips Scattering Theory”, Duke Math. J., 83 (1996), 683–699 | DOI | MR | Zbl
[58] Tr. Mosk. Mat. Obšč., 19 (1968), 211–270 | MR | Zbl
[59] B. Simon, The Statistical Mechanics of Lattice Gases, v. 1, Princeton Univ. Press, Princeton, 1993 | MR
[60] B. Simon, Orthogonal Polynomials on the Unit Circle. Part 1. Classical Theory ; Part 2. Spectral Theory, AMS Colloquium Ser., Providence, RI, 2005 | MR
[61] B. Simon, Trace Ideals and Their Applications, Math. Surv. and Monogr., 120, 2nd ed., AMS, Providence, RI, 2005 | MR | Zbl
[62] K. B. Sinha, A. N. Mohapatra, “Spectral Shift Function and Trace Formula”, Proc. Indian Acad. Sci. (Math. Sci.), 104 (1994), 819–853 | MR | Zbl
[63] E. M. Stein, R. Shakarchi, Fourier Analysis. An Introduction. Princeton Lectures in Analysis, 1, Princeton Univ. Press, Princeton, NJ, 2003 | MR
[64] E. C. Titchmarsh, “On Expansions in Eigenfunctions, VI”, Quart. J. Math. Oxford, 12 (1941), 154–166 | DOI | MR | Zbl
[65] D. Voiculescu, “On a Trace Formula of M. G. Krein”, Operators in Indefinite Metric Spaces, Scattering Theory and Other Topics (Bucharest, 1985), Oper. Theory Adv. Appl., 24, Birkhäuser, Basel, 1987, 329–332 | MR
[66] J. von Neumann, “Charakterisierung des Spektrums eines Integraloperators”, Actualités Sci. Industr., 229 (1935), 3–20
[67] J. von Neumann, Mathematical Foundations of Quantum Mechanics, 12th Printing. Princeton Landmarks in Mathematics, Princeton Paperbacks, Princeton Univ. Press, Princeton, NJ, 1996 | MR | Zbl
[68] H. Weyl, “Über beschränkte quadratische Formen, deren Differenz Vollstetig ist”, Rend. Circ. Mat. Palermo, 27 (1909), 373–392 | DOI | Zbl
[69] D. R. Yafaev, Mathematical Scattering Theory, Transl. Math. Monogr., 105, AMS, Providence, RI, 1992 | MR | Zbl
[70] Funktsional. Anal. i Prilozhen., 41:3 (2007), 60–83 | DOI | MR | Zbl
[71] A. Zygmund, Trigonometric Series, v. I,II, 2nd ed., Cambridge Univ. Press, Cambridge, 1990