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Classes of uniform convergence of spectral expansions for the one-dimensional Schrödinger operator with a distribution potential

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Abstract

For the self-adjoint Schrödinger operator ℒ defined on ℝ by the differential operation −(d/dx)2 + q(x) with a distribution potential q(x) uniformly locally belonging to the space W 2 −1, we describe classes of functions whose spectral expansions corresponding to the operator ℒ absolutely and uniformly converge on the entire line ℝ. We characterize the sharp convergence rate of the spectral expansion of a function using a two-sided estimate obtained in the paper for its generalized Fourier transforms.

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References

  1. Hryniv, R.O. and Mykytyuk, Ya.V., 1-D Schrödinger operators with periodic singular potentials, Methods Funct. Anal. Topology, 2001, vol. 7, no. 4, pp. 31–42.

    MathSciNet  MATH  Google Scholar 

  2. Hryniv, R.O. and Mykytyuk, Ya.V., Self-adjointness of Schrödinger operators with singular potentials, Methods Funct. Anal. Topology, 2012, vol. 18, no. 2, pp. 152–159.

    MathSciNet  MATH  Google Scholar 

  3. Levitan, B.M. and Sargsyan, I.S., Operatory Shturma–Liuvillya i Diraka (Sturm–Liouville and Dirac Operators), Moscow: Nauka, 1988.

    MATH  Google Scholar 

  4. Kostyuchenko, A.G. and Mityagin, B.S., Uniform convergence of spectral resolutions, Funct. Anal. Appl., 1973, vol. 7, no. 2, 113–121.

    Article  MATH  Google Scholar 

  5. Il’in, V.A., Spectral Theory of Differential Operators. Self-Adjoint Differential Operators, New York: Plenum Publ., 1995.

    Book  MATH  Google Scholar 

  6. Il’in, V.A. and Kritskov, L.V., An estimate that is uniform on the whole line for generalized eigenfunctions of the one-dimensional Schrödinger operator with a uniformly locally summable potential, Differ. Equations, 1995, vol. 31, no. 8, pp. 1267–1274.

    MathSciNet  MATH  Google Scholar 

  7. Il’in, V.A. and Antoniu, I., An estimate, uniform over the whole line R, for the deviation of an expanded function from its spectral expansion corresponding to the Schrödinger operator with a bounded and measurable potential, Differ. Equations, 1995, vol. 31, no. 10, pp. 1613–1621.

    MathSciNet  Google Scholar 

  8. Denisov, S.A., An estimate, uniform on the whole line R, for the rate of convergence of a spectral expansion corresponding to the Schrödinger operator with a potential from the Kato class, Differ. Equations, 1997, vol. 33, no. 6, pp. 757–764.

    MathSciNet  MATH  Google Scholar 

  9. Albeverio, S., Gesztesy, F., Høegh-Krohn, R., and Holden, H., Solvable Models in Quantum Mechanics, 2nd ed., Providence, Rhode Island, 2004.

    MATH  Google Scholar 

  10. Savchuk, A.M. and Shkalikov, A.A., Sturm–Liouville operators with singular potentials, Math. Notes, 1999, vol. 66, no. 6, pp. 741–753.

    Article  MathSciNet  MATH  Google Scholar 

  11. Neiman-zade, M.I. and Savchuk, A.M., Schrödinger operators with singular potentials, Proc. Steklov Inst. Math., 2002, vol. 236, pp. 250–259.

    MATH  Google Scholar 

  12. Vladykina, V.E. and Shkalikov, A.A., Asymptotics of the solutions of the Sturm–Liouville equation with singular coefficients, Math. Notes, 2015, vol. 98, no. 5, pp. 891–899.

    MathSciNet  MATH  Google Scholar 

  13. Sadovnichaya, I.V., Equiconvergence of eigenfunction expansions for Sturm–Liouville operators with a distributional potential, Sb. Math., 2010, vol. 201, no. 9, pp. 1307–1322.

    Article  MathSciNet  MATH  Google Scholar 

  14. Kritskov, L.V., Uniform, on the entire axis, convergence of the spectral expansion for Schrödinger operator with a potential-distribution, Differ. Equations, 2017, vol. 53, no. 2, pp. 180–191.

    Article  Google Scholar 

  15. Nikol’skii, S.M., Priblizhenie funktsii mnogikh peremennykh i teoremy vlozheniya (Approximation of Functions of Several Variables and Embedding Theorems), Moscow: Nauka, 1977.

    Google Scholar 

  16. Erdélyi, A., Magnus, W., Oberhettinger, F., and Tricomi, F.G., Higher Transcendental Functions (Bateman Manuscript Project ), New York: McGraw-Hill, 1953. Translated under the title Vysshie transtsendentnye funktsii, Moscow, 1973, vol. 2.

    MATH  Google Scholar 

  17. Schechter, M., Spectra of Partial Differential Operators, Amsterdam–New York–Oxford, 1986.

    MATH  Google Scholar 

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Authors and Affiliations

  1. Lomonosov Moscow State University, Moscow, 119992, Russia

    L. V. Kritskov

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  1. L. V. Kritskov
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Correspondence to L. V. Kritskov.

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Original Russian Text © L.V. Kritskov, 2017, published in Differentsial’nye Uravneniya, 2017, Vol. 53, No. 5, pp. 591–602.

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Kritskov, L.V. Classes of uniform convergence of spectral expansions for the one-dimensional Schrödinger operator with a distribution potential. Diff Equat 53, 583–594 (2017). https://doi.org/10.1134/S0012266117050020

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  • DOI: https://doi.org/10.1134/S0012266117050020

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