Abstract
We study the relationship between the spectral shift function and the excess charge in potential scattering theory. Although these quantities are closely related to each other, they have been often formulated in different settings so far. Here, we first give an alternative construction of the spectral shift function, and then we prove that the spectral shift function thus constructed yields the Friedel sum rule.
Article PDF
Similar content being viewed by others
References
Agmon S.: Spectral properties of Schrödinger operators and scattering theory. Ann. Scuola Norm. Sup. Pisa Cl. Sci. 4 2(2), 151–218 (1975)
Birman M.Sh., Yafaev D.R.: The spectral shift function. The papers of M. G. Kreĭn and their further development. St. Petersburg Math. J. 4(5), 833–870 (1993)
Friedel J.: The distribution of electrons round impurities in monovalent metals. Phil. Mag. 43, 153–189 (1952)
Friedel J.: Electric structure of primary solid solutions in metals. Adv. Phys. 3, 446–507 (1954)
Friedel J.: Structure électronique des impuretés dans les métaux. Ann. Phys. 9, 158–202 (1954)
Friedel J.: Metallic alloys. Nuovo Cimento Suppl. 7, 287–311 (1958)
Gesztesy, F., Nichols, R.: Weak convergence of spectral shift functions for one-dimensional Schrödinger operators. arXiv:1111.0095 (Preprint)
Gesztesy, F., Nichols, R.: An abstract approach to weak convergence of spectral shift functions and applications to multi-dimensional Schrödinger operators. arXiv:1111.0096 (Preprint)
Hislop P.D., Müller P.: The spectral shift function for compactly supported perturbations of Schrödinger operators on large bounded domains. Proc. Am. Math. Soc. 138, 2141–2150 (2010)
Hislop, P.D., Müller, P.: Uniform convergence of spectral shift functions. arXiv:1007.2670 (preprint)
Isozaki H., Kitada H.: A remark on the micro-local resolvent estimates for two body Schrödinger operators. Publ. RIMS 21, 889–910 (1985)
Kato T.: Growth properties of solutions of the reduced wave equation with a variable coefficient. Comm. Pure Appl. Math. 12, 403–425 (1959)
Kirsch W.: Small perturbations and the eigenvalues of the Laplacian on large bounded domains. Proc. Am. Math. Soc. 101, 509–512 (1987)
Kittel C.: Quantum theory of solids, 2nd edn. Wiley, New York (1987)
Kostrykin V., Makarov K.: On Krein’s example. Proc. Am. Math. Soc. 136(6), 2067–2071 (2008)
Krein M.G.: On the trace formula in perturbation theory (Russian). Math. Sbornik N.S. 33(75), 597–626 (1953)
Lifshitz, I.M.: On a problem in perturbation theory. Uspehi Mat. Nauk 7(1), 171–180 (1952) (Russian)
Moure E.: Opérateurs conjugués et propriétés de propagation. Commun. Math. Phys. 91, 279–300 (1983)
Pushnitski A.: The spectral shift function and the invariance principle. J. Funct. Anal. 183(2), 269–320 (2001)
Reed, M., Simon, B.: Methods of Modern Mathematical Physics. vols. I–IV, pp. 1972–1978. Academic Press, New York
Robert, D.: Semiclassical asymptotics for the spectral shift function. In: Differential Operators and Spectral Theory, American Mathematical Society Transl. Ser. 2, vol. 189, pp. 187–203. American Mathematical Society, Providence (1999)
Robert D., Sordoni V.: Trace formulas and Dirichlet-Neumann problems with variable boundary: the scalar case. Helv. Phys. Acta 69(2), 158–176 (1996)
Yafaev, D.R.: Mathematical scattering theory. General theory. Translations of Mathematical Monographs, vol. 105. American Mathematical Society, Providence (1992)
Yafaev, D.R.: Mathematical scattering theory. Analytic theory. Mathematical Surveys and Monographs, vol. 158. American Mathematical Society, Providence (2010)
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Jan Derezinski.
Rights and permissions
About this article
Cite this article
Kohmoto, M., Koma, T. & Nakamura, S. The Spectral Shift Function and the Friedel Sum Rule. Ann. Henri Poincaré 14, 1413–1424 (2013). https://doi.org/10.1007/s00023-012-0219-3
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00023-012-0219-3