Some Schrödinger operators with dense point spectrum
HTML articles powered by AMS MathViewer
- by Barry Simon PDF
- Proc. Amer. Math. Soc. 125 (1997), 203-208
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
- Tadasi Nakayama, On Frobeniusean algebras. I, Ann. of Math. (2) 40 (1939), 611–633. MR 16, DOI 10.2307/1968946
- John D. Dollard and Charles N. Friedman, On strong product integration, J. Functional Analysis 28 (1978), no. 3, 309–354. MR 492656, DOI 10.1016/0022-1236(78)90091-5
- John D. Dollard and Charles N. Friedman, Product integrals and the Schrödinger equation, J. Mathematical Phys. 18 (1977), no. 8, 1598–1607. MR 449253, DOI 10.1063/1.523446
- Michael S. P. Eastham and Hubert Kalf, Schrödinger-type operators with continuous spectra, Research Notes in Mathematics, vol. 65, Pitman (Advanced Publishing Program), Boston, Mass.-London, 1982. MR 667015
- W. A. Harris Jr. and D. A. Lutz, Asymptotic integration of adiabatic oscillators, J. Math. Anal. Appl. 51 (1975), 76–93. MR 369840, DOI 10.1016/0022-247X(75)90142-0
- A. Kiselev, Absolutely continuous spectrum of one-dimensional Schrödinger operators and Jacobi matrices with slowly decreasing potentials, Comm. Math. Phys. (to appear).
- S. N. Naboko, On the dense point spectrum of Schrödinger and Dirac operators, Teoret. Mat. Fiz. 68 (1986), no. 1, 18–28 (Russian, with English summary). MR 875178
- Michael Reed and Barry Simon, Methods of modern mathematical physics. III, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1979. Scattering theory. MR 529429
- J. von Neumann and E.P. Wigner, Über merkwürdige diskrete Eigenwerte, Z. Phys. 30 (1929), 465–467.
Additional Information
- Barry Simon
- Affiliation: Division of Physics, Mathematics, and Astronomy, California Institute of Technology, 253-37, Pasadena, California 91125
- MR Author ID: 189013
- Email: bsimon@caltech.edu
- 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
- © Copyright 1997 Barry Simon
- Journal: Proc. Amer. Math. Soc. 125 (1997), 203-208
- MSC (1991): Primary 34L99, 81Q05
- DOI: https://doi.org/10.1090/S0002-9939-97-03559-4
- MathSciNet review: 1346989