Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Some Schrödinger operators
with dense point spectrum

Author: Barry Simon
Journal: Proc. Amer. Math. Soc. 125 (1997), 203-208
MSC (1991): Primary 34L99, 81Q05
MathSciNet review: 1346989
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Given any sequence $\{E_{n}\}^{\infty }_{n=1}$ of positive energies and any monotone function $g(r)$ on $(0,\infty )$ with $g(0)=1$, $\lim \limits _{r\to \infty } g(r)=\infty $, we can find a potential $V(x)$ on $(-\infty ,\infty )$ such that $\{E_{n}\}^{\infty }_{n=1}$ are eigenvalues of $-\frac {d^{2}}{dx^{2}}+V(x)$ and $|V(x)|\leq (|x|+1)^{-1}g(|x|)$.

References [Enhancements On Off] (What's this?)

  • [1] F. Atkinson, The asymptotic solutions of second order differential equations, Ann. Math. Pura Appl. 37 (1954), 347-378. MR 16:701f
  • [2] J. Dollard and C. Friedman, On strong product integration, J. Funct. Anal. 28 (1978), 309-354. MR 58:11742a
  • [3] -, Product integrals and the Schrödinger equation, J. Math. Phys. 18 (1977), 1598-1607. MR 56:7558
  • [4] M.S.P. Eastham and H. Kalf, Schrödinger-type Operators with Continuous Spectra, Research Notes in Mathematics 65, Pitman Books Ltd., London, 1982. MR 84i:35107
  • [5] W.A. Harris and D.A. Lutz, Asymptotic integration of adiabatic oscillator, J. Math. Anal. Appl. 51 (1975), 76-93. MR 51:6069
  • [6] A. Kiselev, Absolutely continuous spectrum of one-dimensional Schrödinger operators and Jacobi matrices with slowly decreasing potentials, Comm. Math. Phys. (to appear).
  • [7] S.N. Naboko, Dense point spectra of Schrödinger and Dirac operators, Theor.-math. 68 (1986), 18-28. MR 88h:81029
  • [8] M. Reed and B. Simon, Methods of Modern Mathematical Physics, III. Scattering Theory, Academic Press, New York, 1979. MR 80m:81085
  • [9] J. von Neumann and E.P. Wigner, Über merkwürdige diskrete Eigenwerte, Z. Phys. 30 (1929), 465-467.

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (1991): 34L99, 81Q05

Retrieve articles in all journals with MSC (1991): 34L99, 81Q05

Additional Information

Barry Simon
Affiliation: Division of Physics, Mathematics, and Astronomy, California Institute of Technology, 253-37, Pasadena, California 91125

Received by editor(s): July 26, 1995
Additional Notes: This material is based upon work supported by the National Science Foundation under Grant No. DMS-9401491. The Government has certain rights in this material.
Communicated by: Palle E. T. Jorgensen
Article copyright: © Copyright 1997 Barry Simon

American Mathematical Society