On completeness of root vectors of Schrödinger operators: a spectral approach

Volume 10, Issue 1, pp 227--233 http://dx.doi.org/10.22436/jnsa.010.01.22

Publication Date: January 27, 2017 Submission Date: June 06, 2016

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Authors

Xiangdong Yang - Department of Mathematics, KunMing University of Science and Technology, 650093 KunMing, YunNan Province, China.

Abstract

We study complete properties of root vectors of Schrödinger operators. More accurately, denote by \(B(r_0)\) be the ball centered at the origin with radius \(r_0\) and \(L^1(B(r_0))\) the space which consists of real functions f(x) satisfying \(\int_{B(r_0)}|f(x)|dx<\infty\), then the complete properties of eigenvectors for Schrödinger equation are characterized. Our characterization depends on the sum of eigenvalues. Our proof is based on a complex-analytic conjugate approach which is widely used in the investigation of completeness of function systems in Banach spaces.

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Keywords

• Schrödinger operators
• inverse eigenvalue problem
• completeness.

MSC

•  35A99
•  30B60

References

• [1] R. P. Boas, Jr., Entire functions, Academic Press Inc., New York (1954)
  • Google Scholar


• [2] A. Boivin, H.-L. Zhong, Completeness of systems of complex exponentials and the Lambert W functions, Trans. Amer. Math. Soc., 359 (2007), 1829–1849.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [3] T. Bouhennache, Point spectrum of elliptic operators in fibered half-cylinders and the related completeness problem, Integral Equations Operator Theory, 39 (2001), 182–192.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [4] N. Dunford, J. Schwartz, Linear operators, Part I, General theory, With the assistance of William G. Bade and Robert G. Bartle, Reprint of the 1958 original. Wiley Classics Library, A Wiley-Interscience Publication, John Wiley & Sons, Inc., New York (1988)
  • Google Scholar


• [5] R. L. Frank, A. Laptev, E. H. Lieb, R. Seiringer, Lieb-Thirring inequalities for Schrödinger operators with complex-valued potentials, Lett. Math. Phys., 77 (2006), 309–316.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [6] J. B. Garnett, Bounded analytic functions, Pure and Applied Mathematics, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London (1981)
  • Google Scholar


• [7] F. Gesztesy, V. Tkachenko, A Schauder and Riesz basis criterion for non-self-adjoint Schrödinger operators with periodic and antiperiodic boundary conditions, J. Differential Equations, 253 (2012), 400–437.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [8] S. A. Guda, Completeness of the Floquet solutions to the problem of a solid oscillating in a fluid, Sib. Math. J., 50 (2009), 405–414.
  • View Article
  • Google Scholar
  • MATH


• [9] V. Isakov , Completeness of products of solutions and some inverse problems for PDE, J. Differential Equations, 92 (1991), 305–316.
  • View Article
  • MathSciNet
  • Google Scholar


• [10] L. Knockaert, D. D. Zutter, On the completeness of eigenmodes in a parallel plate waveguide with a perfectly matched layer termination, IEEE Trans. Antennas. Propag., 50 (2002), 1650–1653.
  • Google Scholar


• [11] A. Laptev, O. Safronov, Eigenvalue estimates for Schrödinger operators with complex potentials, Comm. Math. Phys., 292 (2009), 29–54.
  • View Article
  • Google Scholar


• [12] G. V. Radzıevksiĭ, Multiple completeness of the root vectors of an M. V. Keldyš pencil that is perturbed by an operator-valued function analytic in the disc, (Russian) Mat. Sb. (N.S.), 91 (1973), 310–335.
  • View Article
  • MathSciNet
  • Google Scholar


• [13] A. G. Ramm, Completeness of the products of solutions to PDE and uniqueness theorems in inverse scattering, Inverse Problems, 3 (1987), 77–82.
  • View Article
  • MathSciNet
  • Google Scholar


• [14] A. G. Ramm, An inverse scattering problem with part of the fixed-energy phase shifts, Comm. Math. Phys., 207 (1999), 231–247.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [15] A. G. Ramm, Uniqueness of the solution to inverse scattering problem with scattering data at a fixed direction of the incident wave, J. Math. Phys., 52 (2011), 12 pages.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [16] Z. Wang, H.-Y. Wu, The completeness of eigenfunctions of perturbation connected with Sturm-Liouville operators, J. Syst. Sci. Complex., 19 (2006), 527–537.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [17] Y. Yakubov, Completeness of elementary solutions of second order elliptic equations in a semi-infinite tube domain, Electron. J. Differential Equations, 2002 (2002), 21 pages.
  • MathSciNet
  • Google Scholar
  • MATH


• [18] X.-D. Yang, Incompleteness of exponential system in the weighted Banach space, J. Approx. Theory, 153 (2008), 73–79.
  • View Article
  • MathSciNet
  • Google Scholar


• [19] X.-D. Yang, Random inverse spectral problems and closed random exponential systems, Inverse Problems, 30 (2014), 12 pages.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [20] X.-D. Yang, J. Tu, On the completeness of the system \(\{t^{\lambda_n}\}\) in \(C_0(E)\), J. Math. Anal. Appl., 368 (2010), 429–437.