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Approximate commutativity for a decaying potential and a function of an elliptic operator


Author: V. A. Sloushch
Translated by: A. V. Kiselev
Original publication: Algebra i Analiz, tom 26 (2014), nomer 5.
Journal: St. Petersburg Math. J. 26 (2015), 849-857
MSC (2010): Primary 35P20
DOI: https://doi.org/10.1090/spmj/1362
Published electronically: July 27, 2015
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Abstract: For a continuous function $ \varphi (\lambda )$, $ \lambda \in \mathbb{R}$, with compact support, a bounded function $ W(x)$, $ x\in \mathbb{R}^{d}$, with power-like asymptotics at infinity, and a suitable selfadjoint operator $ H$ in $ L_{2}({\mathbb{R}}^{d})$, estimates for the singular values of the operator $ \varphi (H)W-W\varphi (H)$ are considered. It is proved that the singular values of $ \varphi (H)W-W\varphi (H)$ decay faster than those of $ \varphi (H)W$. A relationship between the singular values asymptotics for the operators $ \varphi (H)W$ and $ \varphi ^{n}(H)W^{n}$ is also established.


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  • 1. V. A. Sloushch, Cwikel type estimates as a consequence of some properties of the heat kernel, Algebra i Analiz 25 (2013), no. 5, 173-201; English transl., St. Petersburg Math. J. 25 (2014), no. 5, 835-854. MR 3184610
  • 2. S. Alama, P. A. Deift, and R. Hempel, Eigenvalue branches of the Schrödinger operator $ H-\lambda W$ in a gap of $ \sigma (H)$, Commun. Math. Phys. 121 (1989), no. 2, 291-321. MR 985401 (90e:35046)
  • 3. A. B. Pushnitskiĭ and M. V. Ruzhenskiĭ, The spectral shift function of the Schrödinger operator in the large coupling constant limit, Funktsional. Anal. i Prilozen. 36 (2002), no. 3, 93-95; English transl., Funct. Anal. Appl. 36 (2002), no. 3, 250-252. MR 1935913 (2003h:35190)
  • 4. M. Sh. Birman and V. A. Sloushch, Discrete spectrum of the periodic Schrödinger operator with a variable metric perturbed by a nonnegative potential, Math. Model. Nat. Phenom. 5 (2010), no. 4, 32-53, doi:10.1051/mmnp/20105402. MR 2662449 (2011e:47085)
  • 5. M. Sh. Birman and M. Z. Solomyak, Asymptotic behavior of the spectrum of pseudodifferential operators with anisotropically homogeneous symbols. II, Vestnik Leningrad. Univ. Ser. Mat. Mekh. Astronom. 1979, vyp. 3, 5-10; English transl., Vestnik Leningrad. Univ. Math. 12 (1980), 155-161. MR 555971 (81b:47060)
  • 6. D. G. Aronson, Bounds for the fundamental solution of a parabolic equation, Bull. Amer. Math. Soc. 73 (1967), 890-896. MR 0217444 (36:534)
  • 7. E. B. Davies, Heat kernels and spectral theory, Cambridge Tracts in Math., vol. 92, Cambridge Univ. Press, Cambridge, 1989. MR 990239 (90e:35123)
  • 8. M. Sh. Birman and M. Z. Solomyak, Estimates for the singular numbers of integral operators, Uspekhi Mat. Nauk 32 (1977), no. 1, 17-84; English transl., Russian Math. Surveys 32 (1977), no. 1, 15-89. MR 0438186 (55:11104)
  • 9. M. Sh. Birman, G. E. Karadzhov, and M. Z. Solomyak, Boundedness conditions and spectrum estimates for the operators $ b(X)a(D)$ and their analogs, Estimates and Asymptotics for Discrete Spectra of Integral and Differential Equations (Leningrad, 1989-90), Adv. Soviet Math., vol. 7, Amer. Math. Soc., Providence, RI, 1991, pp. 85-106. MR 1306510 (95g:47075)
  • 10. M. Sh. Birman and M. Z. Solomyak, Compact operators with power asymptotic behavior of the singular numbers, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 126 (1983), 21-30; English transl., J. Soviet. Math. 27 (1984), 2442-2447. MR 697420 (85e:47027)
  • 11. -, Spectral theory of selfadjoint operators in Hilbert operators, 2nd ed., revised and augmented, Lan', St. Petersburg, 2010. (Russian)
  • 12. V. A. Sloushch, Generalizations of the Cwikel estimate for integral operators, Tr. S.-Peterb. Mat. Obshch. 14 (2008), 169-196; English transl., Proc. St. Petersburg Math. Soc., vol. 14, Amer. Mat., Providence, RI, 2008, pp. 133-155. MR 2584397 (2011a:47098)

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Additional Information

V. A. Sloushch
Affiliation: Physics Department, St. Petersburg State university, Ul′anovskaya 3, Petrodvorets, St. Petersburg 198504, Russia
Email: vsloushch@list.ru

DOI: https://doi.org/10.1090/spmj/1362
Keywords: Elliptic differential operators, integral operators, estimates for singular values, classes of compact operators
Received by editor(s): March 20, 2014
Published electronically: July 27, 2015
Additional Notes: Supported by RFBR (grant no. 14-01-00760) and by St.Petersburg State University (grant no. 11.38.263.2014).
Article copyright: © Copyright 2015 American Mathematical Society

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