Discrete and Embedded Eigenvalues for One-Dimensional Schrödinger Operators
- 87 Downloads
I present an example of a discrete Schrödinger operator that shows that it is possible to have embedded singular spectrum and, at the same time, discrete eigenvalues that approach the edges of the essential spectrum (much) faster than exponentially. This settles a conjecture of Simon (in the negative). The potential is of von Neumann-Wigner type, with careful navigation around a previously identified borderline situation.
KeywordsNontrivial Solution Asymptotic Formula Essential Spectrum Jacobi Operator Oscillation Theory
Unable to display preview. Download preview PDF.
- 5.Damanik, D., Remling, C.: Schrödinger operators with many bound states. To appear in Duke Math. J.Google Scholar
- 6.Eastham, M.S.P.: The Asymptotic Solution of Linear Differential Systems. Oxford: Oxford University Press (1989)Google Scholar
- 8.Simon, B.: Orthogonal Polynomials on the Unit Circle, Part 2: Spectral Theory. Colloquium Publications 54, Providence, RI: Amer. Math. Soc., (2005)Google Scholar
- 9.Teschl, G.: Jacobi Operators and Completely Integrable Nonlinear Lattices. Mathematical Surveys and Monographs 72, Providence, RI: Amer. Math. Soc., (2000)Google Scholar
- 10.von Neumann J. and Wigner E. (1929). Über merkwürdige diskrete Eigenwerte. Z. Phys. 30: 465–467 Google Scholar