The Schr\"odinger Operator: Perturbation Determinants, the Spectral Shift Function, Trace Identities, and All That

D. R. Yafaev

Funkcionalʹnyj analiz i ego priloženiâ, Tome 41 (2007) no. 3, pp. 60-83.

Résumé

We discuss applications of the M. G. Krein theory of the spectral shift function to the multidimensional Schrödinger operator. Specific properties of this function, for example, its high-energy asymptotics are studied. Trace identities are derived.

Détail

Keywords: M. G. Krein spectral shift function, multidimensional Schrödinger operator, high-energy asymptotics, trace identities.
Mots-clés : trace formula

Bibliographie

[1] S. Agmon and Y. Kannai, “On the asymptotic behavior of spectral functions and resolvent kernels of elliptic operators”, Israel J. Math., 5 (1967), 1–30 | DOI | MR | Zbl

[2] V. M. Babich, Yu. O. Rapoport, “Asimptotika pri malykh vremenakh fundamentalnogo resheniya zadachi Koshi dlya parabolicheskogo uravneniya vtorogo poryadka”, Problemy matem. fiz., 7 (1974), 21–38

[3] M. Sh. Birman, M. G. Krein, “K teorii volnovykh operatorov i operatorov rasseyaniya”, Dokl. AN SSSR, ser. matem., 144 (1962), 475–478 | MR | Zbl

[4] M. Sh. Birman, M. Z. Solomyak, “Zamechaniya o funktsii spektralnogo sdviga”, Zap. nauchn. sem. LOMI, 27 (1972), 33–46 | MR | Zbl

[5] M. Sh. Birman, D. R. Yafaev, “Funktsiya spektralnogo sdviga. Raboty M. G. Kreina i ikh dalneishee razvitie”, Algebra i analiz, 4:5 (1992), 1–44 | MR | Zbl

[6] M. Sh. Birman, D. R. Yafaev, “Spektralnye svoistva matritsy rasseyaniya”, Algebra i analiz, 4:6 (1992), 1–27 | MR | Zbl

[7] J.-M. Bouclet, “Trace formulae for relatively Hilbert–Schmidt perturbations”, Asympt. Anal., 32:3–4 (2002), 257–291 | MR | Zbl

[8] V. S. Buslaev, “Formuly sledov i nekotorye asimptoticheskie otsenki yadra rezolventy dlya operatora Shrëdingera v trekhmernom prostranstve”, Problemy matem. fiz., 1 (1966), 82–101 | MR | Zbl

[9] V. S. Buslaev, L. D. Faddeev, “O formulakh sledov dlya singulyarnogo differentsialnogo operatora Shturma–Liuvillya”, Dokl. AN SSSR, ser. matem., 132 (1960), 13–16 | MR | Zbl

[10] Y. Colin de Verdière, “Une formule de traces pour l'opérateur de Schrödinger dans $\mathbb{R}^3$”, Ann. Sci. École Norm. Sup. (4), 14:1 (1981), 27–39 | DOI | MR | Zbl

[11] I. Ts. Gokhberg, M. G. Krein, Vvedenie v teoriyu lineinykh nesamosopryazhennykh operatorov v gilbertovom prostranstve, Nauka, M., 1965 | MR

[12] V. E. Zakharov, L. D. Faddeev, “Uravnenie Kortevega-de Frisa — vpolne integriruemaya gamiltonova sistema”, Funkts. analiz i ego pril., 5:4 (1971), 18–27 | MR | Zbl

[13] M. Hitrik, I. Polterovich, “Resolvent expansions and trace regularizations for Schrödinger operators”, Contemp. Math., 327 (2003), 161–173 | DOI | MR | Zbl

[14] M. Hitrik, I. Polterovich, “Regularized traces and Taylor expansions for the heat semigroup”, J. London Math. Soc., 68:2 (2003), 402–418 | DOI | MR | Zbl

[15] A. Jensen, T. Kato, “Spectral properties of Schrödinger operators and time-decay of the wave functions”, Duke Math. J., 46:3 (1979), 583–611 | DOI | MR | Zbl

[16] S. Kantorovitz, “$\mathbb{C}^n$-operational calculus, noncommutative Taylor formula and perturbation of semigroups”, J. Funct. Anal., 113:1 (1993), 139–152 | DOI | MR | Zbl

[17] L. S. Koplienko, “K teorii funktsii spektralnogo sdviga”, Problemy matem. fiz., 5 (1971), 62–72 | MR | Zbl

[18] L. S. Koplienko, “Formula sleda dlya vozmuschenii neyadernogo tipa”, Sib. matem. zh., 25:5 (1984), 62–71 | MR | Zbl

[19] M. G. Krein, “O formule sleda v teorii vozmuschenii”, Matem. sb., 33 (1953), 597–626 | MR | Zbl

[20] M. G. Krein, “Ob opredelitelyakh vozmuscheniya i formule sledov dlya unitarnykh i samosopryazhennykh operatorov”, Dokl. AN SSSR, ser. matem., 144 (1962), 268–271 | MR | Zbl

[21] M. G. Krein, “O nekotorykh novykh issledovaniyakh po teorii vozmuschenii”, Pervaya letnyaya matem. shkola, Naukova dumka, Kiev, 1964, 103–187 | MR

[22] S. T. Kuroda, “Scattering theory for differential operators”, J. Math. Soc. Japan, 25:1, 2 (1973), 75–104, 222–234 | DOI | MR | Zbl

[23] I. M. Lifshits, “Ob odnoi zadache teorii vozmuschenii”, UMN, 7:1 (1952), 171–180 | MR | Zbl

[24] R. Newton, “Noncentral potentials: The generalized Levinson theorem and the structure of the spectrum”, J. Math. Phys., 18:7 (1977), 1348–1357 | DOI | MR

[25] V. V. Peller, “Operatory Gankelya v teorii vozmuschenii unitarnykh i samosopryazhennykh operatorov”, Funkts. analiz i ego pril., 19:2 (1985), 37–51 | MR | Zbl

[26] I. Polterovich, “Heat invariants of Riemannian manifolds”, Israel J. Math., 119 (2000), 239–252 | DOI | MR | Zbl

[27] M. Rid, B. Saimon, Metody sovremennoi matematicheskoi fiziki, t. 3, 4, Mir, Moskva, 1982 | MR

[28] D. Robert, “Asymptotique à grande énergie de la phase de diffusion pour un potentiel”, Asympt. Anal., 3:4 (1991), 301–320 | MR

[29] D. Robert, “Semiclassical asymptotics for the spectral shift function”, Differential operators and spectral theory, Amer. Math. Soc. Transl., Ser. 2, 189, Amer. Math. Soc., Providence, RI, 1999, 187–203 | MR | Zbl

[30] M. A. Shubin, Psevdodifferentsialnye operatory i spektralnaya teoriya, Nauka, M., 1978 | MR

[31] M. M. Skriganov, “Ravnomernye koordinatnye i spektralnye asimptotiki reshenii zadachi rasseyaniya dlya uravneniya Shrëdingera”, Zap. nauchn. sem. LOMI, 69 (1977), 171–199 | MR | Zbl

[32] D. R. Yafaev, “Zamechanie o teorii rasseyaniya dlya vozmuschennogo poligarmonicheskogo operatora”, Matem. zametki, 15 (1974), 445–454 | Zbl

[33] D. R. Yafaev, Matematicheskaya teoriya rasseyaniya, Izd-vo S.-Peterburgskogo Universiteta, 1994 | MR

[34] D. R. Yafaev, “High energy asymptotics of the scattering amplitude for the Schrödinger equation”, Proc. Indian Acad. Sci., Math. Sci., 112:1 (2002), 245–255 | DOI | MR | Zbl

[35] D. R. Yafaev, “High energy and smoothness asymptotic expansion of the scattering amplitude”, J. Funct. Anal., 202:2 (2003), 526–570 | DOI | MR | Zbl