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The Schrödinger operator: Perturbation determinants, the spectral shift function, trace identities, and all that. (English) Zbl 1222.47072

Funct. Anal. Appl. 41, No. 3, 217-236 (2007); translation from Funkts. Anal. Prilozh. 41, No. 3, 60-83 (2007).
The author studies a new approach to the construction of an asymptotic expansion of the function \(\xi(\lambda)\) for \(\lambda\to\infty\). The existence of such an expansion is shown using the formula of Birman-Krein (higher energetical asymptotic expansion of \(S(\lambda)\)). The coefficients are effectively calculated. Moreover, a trace identity is given for entire and half orders as well as characterised in terms of invariants of the heat equation. The whole paper is based on Krein’s theory applied to the Schrödinger operator.

MSC:

47F05 General theory of partial differential operators
35P20 Asymptotic distributions of eigenvalues in context of PDEs
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[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. · Zbl 0148.13003
[2] V. M. Babich and Yu. O. Rapoport, ”The short-time asymptotic behavior of the fundamental solution of the Cauchy problem for the parabolic equation of the second order,” in: Problemy Matem. Fiz., vol. 7, 1974, 21–38.
[3] M. Sh. Birman and M. G. Krein, ”On the theory of wave operators and scattering operators,” Dokl. Akad. Nauk SSSR, Ser. Mat., 144 (1962), 475–478; English transl.: Soviet Math. Dokl., 3 (1962), 740–744.
[4] M. Sh. Birman and M. Z. Solomyak, ”Remarks on the spectral shift function,” Zap. Nauchn. Sem. LOMI, 27 (1972), 33–46; English transl.: J. Soviet Math., 3:4 (1975). · Zbl 0329.47009
[5] M. Sh. Birman and D. R. Yafaev, ”The spectral shift function. The papers of M. G. Krein and their further development,” Algebra i Analiz, 4:5 (1992), 1–44; English transl.: St. Petersburg Math. J., 4:5 (1993), 833–870.
[6] M. Sh. Birman and D. R. Yafaev, ”Spectral properties of the scattering matrix,” Algebra i Analiz, 4:6 (1992), 1–27; English transl.: St. Petersburg Math. J., 4:6 (1993), 1055–1079. · Zbl 0819.47009
[7] J.-M. Bouclet, ”Trace formulae for relatively Hilbert-Schmidt perturbations,” Asympt. Anal., 32:3–4 (2002), 257–291. · Zbl 1062.47021
[8] V. S. Buslaev, ”Trace formulas and some asymptotic estimates of the resolvent kernel of the Schrödinger operator in dimension three,” in: Problemy Matem. Fiz., 1966, 82–101; English transl.: in Topics in Math. Physics, vol. 1, Plenum Press, 1966. · Zbl 0203.13403
[9] V. S. Buslaev and L. D. Faddeev, ”Formulas for traces for a singular Sturm-Liouville differential operator,” Dokl. Akad. Nauk SSSR, Ser. Matem., 132 (1960), 13–16; English transl.: Soviet Math. Dokl., 1 (1960), 451–454. · Zbl 0129.06501
[10] Y. Colin de Verdière, ”Une formule de traces pour l’opérateur de Schrö dinger dans \(\mathbb{R}\)3,” Ann. Sci. École Norm. Sup. (4), 14:1 (1981), 27–39. · Zbl 0482.35068
[11] I. C. Gokhberg and M. G. Krein, Introduction to the theory of linear nonselfadjoint operators, Amer. Math. Soc., Providence, R.I., 1969. · Zbl 0181.13504
[12] V. E. Zakharov and L. D. Faddeev, ”Korteweg-de Vries equation: A completely integrable Hamiltonian system,” Funkts. Anal. Prilozhen., 5:4 (1971), 18–27; English transl.: Funct. Anal. Appl., 5 (1971), 280–287. · Zbl 0257.35074
[13] M. Hitrik and I. Polterovich, ”Resolvent expansions and trace regularizations for Schrödinger operators,” Contemp. Math., 327 (2003), 161–173. · Zbl 1109.35030
[14] M. Hitrik and I. Polterovich, ”Regularized traces and Taylor expansions for the heat semigroup,” J. London Math. Soc., 68:2 (2003), 402–418. · Zbl 1167.35335
[15] A. Jensen and T. Kato, ”Spectral properties of Schrödinger operators and time-decay of the wave functions,” Duke Math. J., 46:3 (1979), 583–611. · Zbl 0448.35080
[16] S. Kantorovitz, ”\(\mathbb{C}\)n-operational calculus, noncommutative Taylor formula and perturbation of semigroups,” J. Funct. Anal., 113:1 (1993), 139–152. · Zbl 0807.46063
[17] L. S. Koplienko, ”On the theory of the spectral shift function,” in: Problemy Matem. Fiz., vol. 5, 1972, 62–72; English transl.: in Topics in Math. Phys., vol. 5, Plenum Press, New York, 1972. · Zbl 0268.47022
[18] L. S. Koplienko, ”The trace formula for perturbations of non-trace class type,” Sibirsk. Mat. Zh., 25 (1984), 62–71; English transl.: Siberian Math. J., 25:4 (1984), 735–743. · Zbl 0574.47021
[19] M. G. Krein, ”On the trace formula in perturbation theory,” Mat. Sb., 33 (1953), 597–626. · Zbl 0052.12303
[20] M. G. Krein, ”On perturbation determinants and the trace formula for unitary and selfadjoint operators,” Dokl. Akad. Nauk SSSR, Ser. Matem., 144 (1962), 268–271; English transl.: Soviet Math. Dokl., 3 (1962), 707–710.
[21] M. G. Krein, ”Some new studies in the theory of perturbations of selfadjoint operators,” in: Topics in differential and integral equations and operator theory, Birkhäser, Basel, 1983, 107–172.
[22] S. T. Kuroda, ”Scattering theory for differential operators,” J. Math. Soc. Japan, 25:1, 2 (1973), 75–104, 222–234. · Zbl 0245.47006
[23] I. M. Lifshits, ”On a problem in perturbation theory,” Usp. Mat. Nauk, 7:1 (1952), 171–180. · Zbl 0046.21203
[24] R. Newton, ”Noncentral potentials: The generalized Levinson theorem and the structure of the spectrum,” J. Math. Phys., 18:7 (1977), 1348–1357.
[25] V. V. Peller, ”Hankel operators in the perturbation theory of unitary and self-adjoint operators,” Funkts. Anal. Prilozhen., 19:2 (1985), 37–51; English transl.: Funkt. Anal. Appl., 19:2 (1985), 111–123. · Zbl 0587.47016
[26] I. Polterovich, ”Heat invariants of Riemannian manifolds,” Israel J. Math., 119 (2000), 239–252. · Zbl 0996.58019
[27] M. Reed and B. Simon, Methods of modern mathematical physics, vol. 3, 4, Academic Press, San Diego, CA, 1979, 1978. · Zbl 0401.47001
[28] D. Robert, ”Asymptotique à grande énergie de la phase de diffusion pour un potentiel,” Asympt. Anal., 3:4 (1991), 301–320. · Zbl 0737.35054
[29] D. Robert, ”Semiclassical asymptotics for the spectral shift function,” in: Differential operators and spectral theory, Amer. Math. Soc. Transl., Ser. 2, vol. 189, Amer. Math. Soc., Providence, RI, 1999, 187–203. · Zbl 0922.35108
[30] M. A. Shubin, Pseudodifferential Operators and Spectral Theory, Springer-Verlag, 1987. · Zbl 0616.47040
[31] M. M. Skriganov, ”Uniform coordinate and spectral asymptotics for solutions of the scattering problem for the Schrödinger equation,” Zap. Nauchn. Sem. LOMI, 69 (1977), 171–199; English transl.: J. Soviet Math., 10 (1978), 120–141. · Zbl 0359.35015
[32] D. R. Yafaev, ”Remark on scattering theory for a perturbed polyharmonic operator,” Mat. Zametki, 15 (1974), 445–454; English transl.: Math. Notes, 15 (1974), 260–265. · Zbl 0302.47007
[33] D. R. Yafaev, Mathematical Scattering Theory, Amer. Math. Soc., Providence, R.I., 1992. · Zbl 0761.47001
[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. · Zbl 1199.81037
[35] D. R. Yafaev, ”High energy and smoothness asymptotic expansion of the scattering amplitude,” J. Funct. Anal., 202:2 (2003), 526–570. · Zbl 1045.35059
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